A tour of representation theory [draft ed.] 111-111-111-1

Citation preview

Martin Lorenz

A Tour of Representation Theory (preliminary version) January 11, 2017

Contents

Notations and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .XVII Part I Algebras 1

Representations of Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 The Category of k-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Some Important Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Endomorphism Algebras and Matrices . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 The Category Rep A and First Examples . . . . . . . . . . . . . . . . . 1.2.2 Changing the Algebra or the Base Field . . . . . . . . . . . . . . . . . 1.2.3 Irreducible Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Composition Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Endomorphism Algebras and Schur’s Lemma . . . . . . . . . . . . 1.2.6 Indecomposable Representations . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Primitive Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Degree-1 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Commutative Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Connections with Prime and Maximal Ideals . . . . . . . . . . . . . 1.3.4 The Jacobson-Zariski Topology . . . . . . . . . . . . . . . . . . . . . . . . 1.3.5 The Jacobson Radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Semisimplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Completely Reducible Representations . . . . . . . . . . . . . . . . . . 1.4.2 Socle and Homogeneous Components . . . . . . . . . . . . . . . . . . . 1.4.3 Semisimple Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 3 5 12 15 18 23 24 26 29 31 34 36 37 40 40 41 42 43 46 47 49 49 52 54

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1.4.4 Some Consequences of Wedderburn’s Structure Theorem . . 1.4.5 Finite-Dimensional Irreducible Representations . . . . . . . . . . . 1.4.6 Semisimplicity and the Jacobson radical . . . . . . . . . . . . . . . . . 1.4.7 Absolutely Irreducible Representations . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Spaces of Trace Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Irreducible Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.4 The Grothendieck Group R(A) . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 58 60 61 61 63 64 65 68 69 73

Further Topics on Algebras and their Representations . . . . . . . . . . . . . 2.1 Projectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Hattori-Stallings Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 The Grothendieck Groups K0 (A) and P(A) . . . . . . . . . . . . . 2.1.4 Finite-Dimensional Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Frobenius and Symmetric Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Definition of Frobenius and Symmetric Algebras . . . . . . . . . . 2.2.2 Frobenius Form, Dual Bases and Nakayama Automorphism 2.2.3 Casimir Elements, Casimir Operator and Higman Trace . . . . 2.2.4 Trace Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Symmetric Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 Semisimple Algebras as Symmetric Algebras . . . . . . . . . . . . . 2.2.7 Integrality and Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.8 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 77 77 80 82 86 92 94 94 95 98 99 100 102 103 105 106

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Part II Groups 3

Groups and Group Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Group Algebras and Representations of Groups: Generalities . . . . . . 3.1.1 Group Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Representations of Groups and Group Algebras . . . . . . . . . . . 3.1.3 Changing the Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Characters of Finite-Dimensional Group Representations . . . 3.1.5 Finite Group Algebras as Symmetric Algebras . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 First Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Finite Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Degree-1 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 The Dihedral Group D4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 111 111 114 115 118 121 121 122 122 123 124

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3.2.4 Some Representations of the Symmetric Group Sn . . . . . . . . 3.2.5 Permutation Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . More Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Comultiplication, Counit, and Antipode . . . . . . . . . . . . . . . . . 3.3.2 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 A Plethora of Representations . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Characters and Symmetric Polynomials . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Semisimple Group Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 The Semisimplicity Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 The Orthogonality Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 The Case of the Complex Numbers . . . . . . . . . . . . . . . . . . . . . 3.4.4 Primitive Central Idempotents of Group Algebras . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Applications to Invariant Theory . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Symmetric and Antisymmetric Tensors . . . . . . . . . . . . . . . . . . 3.5.2 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Molien’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Exterior Powers of the Standard Representation of Sn . . . . . 3.6.2 The Groups S4 and S5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 The Alternating Groups A4 and A5 . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Classical Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Divisibility Theorems of Frobenius, Schur and Itô . . . . . . . . . 3.7.2 Burnside’s pa q b -Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.3 The Brauer-Fowler Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.4 Clifford Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

126 127 129 129 129 131 133 137 141 144 144 145 148 148 150 153 153 157 158 160 162 162 165 167 170 171 171 173 175 177 179

The Symmetric Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Gelfand-Zetlin Algebras and Jucys-Murphy Elements . . . . . . . . . . . . 4.1.1 Centralizer Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Generators of the Gelfand-Zetlin Algebra . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Branching Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Restricting Irreducible Representations . . . . . . . . . . . . . . . . . . 4.2.2 The Graph B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Gelfand-Zetlin Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Maximality of GZ n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Young Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Partitions and Young Diagrams . . . . . . . . . . . . . . . . . . . . . . . . .

181 182 183 184 185 185 186 186 187 188 189 190 190

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3.5

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4.3.2 The graph Y and the Graph Isomorphism Theorem . . . . . . . . 4.3.3 Some Consequences of the Graph Isomorphism Theorem . . 4.3.4 Paths in Y and Standard Young Tableaux . . . . . . . . . . . . . . . . . 4.3.5 The Hook-Length Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Proof of the Graph Isomorphism Theorem . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Cont(n) = Spec(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 The Irreducible Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Realization over Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Young’s Orthogonal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Skew Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.4 Representations Associated to Skew Shapes . . . . . . . . . . . . . . 4.5.5 The Murnaghan-Nakayama Rule: Statement and Examples . 4.5.6 Proof of the Murnaghan-Nakayama Rule . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Schur-Weyl Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 The Double Centralizer Theorem for Semisimple Algebras . 4.6.2 The Double Centralizer Theorem for Sn and GL(V ) . . . . . . 4.6.3 Schur Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

191 192 193 194 197 198 198 203 206 208 209 209 212 213 215 218 221 226 226 227 229 230 234

Part III Lie Algebras 5

Lie Algebras and Enveloping Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Basics about Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Lie Algebras and their Homomorphisms . . . . . . . . . . . . . . . . . 5.1.2 General Linear Lie Algebras and Representations of Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Subalgebras and Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Examples of Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.5 Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Types of Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Nilpotent and Solvable Lie Algebras . . . . . . . . . . . . . . . . . . . . 5.2.2 Simple and Semisimple Lie Algebras . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Three Theorems about Linear Lie Algebras . . . . . . . . . . . . . . . . . . . . . 5.3.1 Engel’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Lie’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Cartan’s Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

237 238 238 239 239 241 242 244 246 246 248 249 249 249 251 253 256

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Enveloping Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 The Enveloping Algebra of a Lie Algebra . . . . . . . . . . . . . . . . 5.4.2 The Poincaré-Birkhoff-Witt Theorem . . . . . . . . . . . . . . . . . . . 5.4.3 More Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Generalities on Representations of Lie Algebras . . . . . . . . . . . . . . . . . 5.5.1 Invariants and the Trivial Representation . . . . . . . . . . . . . . . . . 5.5.2 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 Tensor, Exterior and Symmetric Powers . . . . . . . . . . . . . . . . . 5.5.5 g-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.6 Adjoint Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.7 The Representation Ring of a Lie Algebra . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 The Nullstellensatz for Enveloping Algebras . . . . . . . . . . . . . . . . . . . . 5.6.1 The Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 The Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 Locally Closed Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.4 Central Characters and Prim Ug for Nilpotent g . . . . . . . . . . 5.6.5 Rational Ideals and the Dixmier-Mœglin Equivalence . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Representations of sl2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 The Representations V (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.2 Finite-Dimensional Representations of sl2 . . . . . . . . . . . . . . . 5.7.3 Weight Ladders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.4 The Casimir Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.5 Proof of Theorem 5.37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.6 Formal Characters and the Representation Ring of sl2 . . . . . 5.7.7 The Center of U(sl2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.8 Prime and Primitive Ideals of U(sl2 ) . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

257 257 259 264 267 269 269 270 271 271 272 273 275 276 277 277 278 280 281 283 289 290 290 292 293 294 295 296 299 299 302

Semisimple Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Characterizations of Semisimplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 The Killing Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Start of the Proof of Theorem 6.1 . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Some Consequences of Theorem 6.1 . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Complete Reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Casimir Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Proof of Weyl’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Abstract Jordan Decomposition . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Cartan Subalgebras and the Root Space Decomposition . . . . . . . . . . . 6.3.1 Cartan Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

305 305 306 307 308 309 309 309 311 312 313 314 314

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6.3.2 Root Space Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Simplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Embedding into Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Classical Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Checking for Semisimplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 The Special Linear Lie Algebra sln+1 (Type An ) . . . . . . . . . . 6.4.3 The Classical Lie Algebras of Types Bn , Cn and Dn . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

316 319 320 322 322 323 323 326 329

7

Root Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Abstract Root Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 The Crystallographic Restriction . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Root Systems of Ranks ≤ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Automorphism Group and Weyl Group . . . . . . . . . . . . . . . . . . 7.1.4 The Classical Root Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Bases of a Root System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Examples: The Classical Root Systems . . . . . . . . . . . . . . . . . . 7.2.2 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Cartan Matrix and Dynkin Diagram . . . . . . . . . . . . . . . . . . . . . 7.3.3 Classification Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Lattices Associated to a Root System . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Root and Weight Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Dominant Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 The Action of the Weyl Group . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.4 Multiplicative Invariants of Weight Lattices . . . . . . . . . . . . . . 7.4.5 Anti-invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

331 331 332 333 333 335 339 339 339 341 342 345 345 345 347 347 349 350 350 352 353 354 356 358

8

Representations of Semisimple Lie Algebras . . . . . . . . . . . . . . . . . . . . . . 8.1 Reminders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 The Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Triangular Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 The Case of sl2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Finite-Dimensional Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Weight Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Weights of Finite-Dimensional Representations . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

359 359 359 361 361 362 362 362 363 364

6.4

Contents

8.3

Finite-Dimensional Irreducible Representations . . . . . . . . . . . . . . . . . 8.3.1 Highest Weight Representations . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Verma Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Fundamental Representations of sln+1 . . . . . . . . . . . . . . . . . . 8.3.5 Weights and Weight Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 The Representation Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Group Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Formal Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Ring Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 Fundamental Characters and Representation Ring of sln+1 . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 The Center of the Enveloping Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Invariant Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Elementary Automorphisms of g . . . . . . . . . . . . . . . . . . . . . . . 8.5.3 Invariants in the Symmetric Algebra . . . . . . . . . . . . . . . . . . . . 8.5.4 Central Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.5 The Harish-Chandra Homomorphism . . . . . . . . . . . . . . . . . . . 8.5.6 The Harish-Chandra Isomorphism . . . . . . . . . . . . . . . . . . . . . . 8.5.7 The Shephard-Todd-Chevalley Theorem . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Weyl’s Character Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 Characters and Dimensions for sln+1 . . . . . . . . . . . . . . . . . . . 8.6.3 Formal Characters Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.4 Proof of Weyl’s Character Formula . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Representations of sl(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1 The Action of gl(V ) on V ⊗n . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.2 Schur Functors Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.3 Sλ V as a Highest Weight Representation . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XI

364 364 365 368 370 371 371 372 372 373 373 374 375 375 375 377 379 382 382 383 386 387 387 388 390 394 395 397 398 398 399 400 402

Part IV Hopf Algebras 9

Coalgebras, Bialgebras and Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Coalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 The Category of k-Coalgebras . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Initial Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Sweedler and Graphical Notation . . . . . . . . . . . . . . . . . . . . . . . 9.1.4 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.5 From Coalgebras to Algebras and Back . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

405 405 405 407 408 412 414 417

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9.2

Comodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Comodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Local Finiteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 The Passage between Comodules and Modules . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bialgebras and Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Bialgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Further Examples of Bialgebras . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.4 Properties of the Antipode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.5 Duals of Bialgebras and Hopf Algebras . . . . . . . . . . . . . . . . . . 9.3.6 Some Examples of Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

419 419 421 422 424 425 425 427 429 430 431 434 438

10

Representations and Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Representations of Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 General Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Hopf Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 First Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Finiteness of Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Some Properties of Finite-Dimensional Hopf Algebras . . . . . 10.2.3 Inner Faithful Representations . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.4 Non-Divisibility Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.5 The Chevalley Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 The Representation Ring of a Hopf Algebra . . . . . . . . . . . . . . . . . . . . 10.3.1 Ring Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 The Character Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 The Representation Ring of a Finite Group Algebra . . . . . . . 10.3.4 Additional Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Actions and Coactions of Hopf Algebras on Algebras . . . . . . . . . . . . 10.4.1 Module and Comodule Algebras . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 Adjoint Action and Chevalley Property . . . . . . . . . . . . . . . . . . 10.4.4 Finite Generation of Invariants . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

441 441 441 447 449 451 451 453 454 456 458 459 460 460 461 462 464 466 467 467 469 471 472 475

11

Affine Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Affine Group Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Group Functors from Hopf Algebras . . . . . . . . . . . . . . . . . . . . 11.1.2 Affine Group Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.3 Categorical View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.4 Some Examples of Affine Group Schemes . . . . . . . . . . . . . . .

477 477 477 478 479 479

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Contents

12

XIII

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Affine Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 The Category of Affine Algebraic k-Groups . . . . . . . . . . . . . . 11.2.3 First Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Representations and Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Rational Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Rational Actions on Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 The Zariski Topology of an Affine Algebraic Group . . . . . . . 11.4.2 Linear Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.3 The Closure of a Linear Group . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Irreducibility and Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.1 Primes and Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.3 The Identity Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Chevalley’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.1 The Lie Algebra of an Affine Algebraic Group . . . . . . . . . . . . 11.6.2 Reductive Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.3 Complete Reducibility of Tensor Products . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Algebraic Group Actions on Prime Spectra . . . . . . . . . . . . . . . . . . . . . 11.7.1 G-Cores and G-Prime Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7.2 A Stratification of the Prime Spectrum . . . . . . . . . . . . . . . . . . 11.7.3 An Example: The Primes of Quantum Affine Space . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

481 482 482 483 484 485 486 486 487 488 488 488 489 490 491 492 492 493 494 495 495 495 496 497 498 498 498 500 500 503

Finite-Dimensional Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Frobenius Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.2 Modular Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.3 Frobenius Form and Casimir Element . . . . . . . . . . . . . . . . . . . 12.1.4 The Nakayama Automorphism . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.5 Symmetric Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 The Antipode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Order of the Antipode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 The Trace of S2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Semisimplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 The Generalized Maschke Theorem . . . . . . . . . . . . . . . . . . . . .

505 505 505 507 508 510 511 511 513 513 514 515 515 515

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12.3.2 Involutory Semisimple Hopf Algebras . . . . . . . . . . . . . . . . . . . 12.3.3 The Representation Algebra of a Semisimple Hopf Algebra . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Divisibility Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Frobenius Divisibility for Hopf Algebras . . . . . . . . . . . . . . . . 12.4.2 Characters that are Central in H ∗ . . . . . . . . . . . . . . . . . . . . . . . 12.4.3 The Class Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.4 Some Applications of the Class Equation . . . . . . . . . . . . . . . . 12.4.5 Freeness over Hopf Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Frobenius-Schur Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.1 Higher Frobenius-Schur Indicators . . . . . . . . . . . . . . . . . . . . . . 12.5.2 The Second Frobenius-Schur Indicator . . . . . . . . . . . . . . . . . . 12.5.3 Return to Finite Group Algebras . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

517 519 521 521 522 523 523 524 526 528 528 529 530 531 533

Part V Appendices A

The Language of Categories and Functors . . . . . . . . . . . . . . . . . . . . . . . . A.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.1 Some Examples of Categories . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.1 First Examples of Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.2 Contravariant Functors and Opposite Categories . . . . . . . . . . A.3 Naturality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.1 Natural Transformations and Functoriality . . . . . . . . . . . . . . . A.3.2 Natural Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.3 Equivalence of Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Adjointness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

537 537 538 539 540 540 541 541 542 543 544

B

Background from Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1.1 The Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1.2 Additional Structure from Bimodules . . . . . . . . . . . . . . . . . . . B.1.3 Commutative Rings: Tensor Powers and Multilinear Maps . . B.2 Hom-⊗ Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2.1 The Hom-Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2.2 Hom-⊗ Adjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3.1 The Bifunctor ⊗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3.2 The Bifunctor Homk and the Linear Dual . . . . . . . . . . . . . . . . B.3.3 The Trace Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3.4 Extensions of the Base Field . . . . . . . . . . . . . . . . . . . . . . . . . . .

547 547 547 549 550 553 553 554 554 554 555 556 557

Contents

XV

C

Some Commutative Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1 The Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 The Generic Flatness Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.3 The Zariski Topology on a Vector Space . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

559 559 560 561 562

D

The Diamond Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.1 The Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.2 The Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.3 First Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.4 A Simplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.5 The Poincaré-Birkhoff-Witt Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

565 565 566 568 570 571 573

E

The Symmetric Ring of Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.2 The Extended Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.3 Comparison with Other Rings of Quotients . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

575 575 577 579 580

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

Notations and Conventions

Functions and actions will be written on the left. In particular, all modules are left modules unless stated otherwise. Rings need not be commutative, but we assume throughout this book that every ring R has an identity element, denoted by 1R or simply 1, and that all ring homomorphisms are unital, that is, the image of the identity element of the source ring is the identity element of the target ring. Ideals in a ring are understood to be two-sided. Throughout, k will denote a commutative base field. Any specific assumptions on k will be explicitly stated, usually at the beginning of each chapter. Here is a list of the main abbreviations and symbols used in the text. General F XI X (I) Z+ , R+ , . . . N [n] Fp k µN

disjoint union of sets the set of functions I → X the set of functions I → X having finite support the non-negative integers, reals, . . . the set of natural numbers, {1, 2, . . . } the set {1, 2, . . . , n} for n ∈ N the field with p elements the base field the group of N th roots of unity in some algebraic closure of k – or the functor

Categories Sets Algk Vectk Vect∆ k

k-algebras k-vector spaces ∆-graded k-vector spaces (∆ some set, often a monoid)

XVIII

Notations and Conventions

Groups Rep A A Mod ModA Repfin A A Proj A proj Projfin A

— etc many more – – – – – – –

Vector spaces V ∗ = Homk (V, k) h·, ·i TV, Tn V = V ⊗n Sym V, Symn V ΛV, Λn V O(V ) = Sym(V ∗ ) Vλ , Vφ kX or k[X] GL(V ) GLn (k) V ⊕I

the dual space of the k-vector space V the evaluation form V ∗ × V → k the tensor algebra of V and its nth component the symmetric algebra of V and its nth component the exterior algebra of V and its nth component the algebra of polynomial functions on V eigenspace, space of semi-invariants, weight space the vector space of all formal linear combinations of a set X the group of invertible linear endomorphisms of V the group of invertible n × n-matrices I-fold direct sum

Algebras Aop A× Irr A Z A or Z (A) MaxSpec A Spec A Prim A

the opposite algebra of A the group of units (invertible elements) of A a full representative set of the isomorphism classes of all irreps of A the center of the algebra A

the set of

Groups kG or k[G] Cn Dn Sn An G X

the group algebra of the group G over k the cyclic group of order n the dihedral group of order 2n the symmetric group on {1, 2, . . . , n} the alternating subgroup of Sn short for G × X → X, a left action of G on the set X

Notations and Conventions

G\X G/H G

x

XIX

the set of orbits for an action G X; often, the same notation will also denote a transversal for these orbits the collection of all left cosets gH (g ∈ G) of a subgroup H ≤ G or a transversal for these cosets the G-conjugacy class of an element x ∈ G

Symmetric Groups Irr Sn GZ n dn Zn = Z (kSn ) fλ

Irr kSn ; the base field k is understood to be algebraically closed of characteristic 0 the Gelfand-Zetlin subalgebra of kSn the dimension of GZ n , equal to the sum of the dimensions of all Sn -irreps the center of the group algebra kSn the number of standard Young tableaux of shape λ

Lie Algebras Ug M (λ) V (λ)

the enveloping algebra of g Verma module the unique irreducible image of M (λ)

Part I

Algebras

1 Representations of Algebras

This chapter develops the basic themes of representation theory in the setting of algebras. We establish notation to be used throughout the remainder of the book and prove some fundamental results of representation theory such as Wedderburn’s Structure Theorem. The focus will be on irreducible and completely reducible representations. We assume that the reader is familiar with basic linear algebra including tensor products — some background is provided in Appendix B — and with the rudiments of the theory of rings and modules. Throughout, k denotes an arbitrary field and Vectk is the category of k-vector spaces and k-linear maps; it is understood that all k-vector spaces have identical left and right scalar operations. Finally, ⊗ will stand for ⊗k .

1.1 Algebras In the language of rings, a k-algebra can be defined as a ring A (with 1) together with a given ring homomorphism k → A that has image in the center Z A of A . In this section, we will first recast this definition in an equivalent form, starting over from scratch in the setting of k-vector spaces rather than rings. We will then proceed to deploy a selection of algebras that will all play prominent roles later on in this book and discuss their main features, taking the opportunity to mention some standard concepts from the theory of algebras and from category theory along the way. 1.1.1 The Category of k-Algebras Working in Vectk at the outset, a k-algebra A can equivalently be defined as a k-vector space that is equipped with two k-linear maps, the multiplication m = mA : A ⊗ A → A and the unit u = uA : k → A, such that the following diagrams commute:

4

1 Representations of Algebras

A⊗A A⊗A⊗A

m ⊗ Id

Id ⊗m

A⊗A

u ⊗ Id

and

m

A⊗A

m

Id ⊗u

k⊗A

A⊗k

m

∼

(1.1)

∼

A A

∼ Here Id = IdA denotes the identity map of A. The isomorphism k ⊗ A −→ A in (1.1) is the standard one, given by the scalar multiplication, λ ⊗ a 7→ λa for λ ∈ k ∼ and a ∈ A; similarly for A ⊗ k −→ A. Multiplication will generally be written as m(a ⊗ b) = ab for a, b ∈ A. Thus, ab depends k-linearly on both a and b. The algebra A is said to be commutative if ab = ba holds for all a, b ∈ A or, equivalently, m = m ◦ τ where τ ∈ Endk (A ⊗ A) is given by τ (a ⊗ b) = b ⊗ a. Commutativity of the first diagram in (1.1) amounts to the associative law,

(ab)c = a(bc) for all a, b, c ∈ A. The second diagram in (1.1) implies that u(1k ) a = a = a u(1k ) for all a ∈ A; so u(1k ) = 1A is the identity element of A. If u = 0, then it follows that A = {0}; otherwise the unit map u is injective and it is often notationally suppressed, viewing it as an inclusion k ⊆ A. Then 1A = 1k and the scalar operation of k on A becomes multiplication in A — so the use of the juxtaposition notation for both is consistent — and k ⊆ Z A. Given k-algebras A and B, a homomorphism from A to B is a k-linear map f : A → B that respects multiplications and units in the sense that the following diagrams commute: A⊗A

f ⊗f

mA

f

B⊗B mB

A

(1.2) uA

f

A

B

and uB

k

B

These diagrams can be equivalently expressed by the equations f (aa0 ) = f (a)f (a0 ) for all a, a0 ∈ A and f (1A ) = 1B . The category whose objects are the k-algebras and whose morphisms are the homomorphisms between k-algebras will be denoted by Algk and HomAlgk (A, B) will denote the set of all k-algebra homomorphisms A → B. We refer to Appendix A for a brief introduction to the language of categories. Algebra homomorphisms are often simply called algebra maps. The variants isomorphism, monomorphism, . . . all have their usual meaning, and the notion of a subalgebra of a given k-algebra A is the obvious one: it is a k-subspace B of A that is a k-algebra in its own right such that the inclusion map B ,→ A is a map of k-algebras.

1.1 Algebras

5

Extending the base field For any A ∈ Algk and any field extension K/k, we may regard K ⊗ A as a K-vector space as in §B.3.4. Together with K ⊗ uA : K ∼ = K ⊗ k → K ⊗ A as unit map and with multiplication K ⊗ mA : (K ⊗ A) ⊗K (K ⊗ A) ∼ = A ⊗ (K ⊗K K) ⊗ A ∼ = K ⊗ (A ⊗ A) → K ⊗ A, we obtain a K-algebra. A more down-to-earth expression for the multiplication of K ⊗ A is (λ ⊗ a)(λ0 ⊗ a0 ) = λλ0 ⊗ aa0 for λ, λ0 ∈ K and a, a0 ∈ A. This yields the field extension functor K ⊗ · : Algk −→ AlgK Tensor products of algebras The tensor product of two algebras A, B ∈ Algk is obtained by endowing the kvector space A ⊗ B with the multiplication (a ⊗ b)(a0 ⊗ b0 ) := aa0 ⊗ bb0

(1.3)

for a, a0 ∈ A and b, b0 ∈ B. It is trivial to check that this multiplication is welldefined and, together with the unit map uA ⊗ uB : k ∼ = k ⊗ k → A ⊗ B, turns A ⊗ B into a k-algebra. Evidently, A⊗B ∼ B ⊗A in Alg via the map a⊗b ↔ b⊗a. In the = k special case where B = K is a field, we obtain the algebra K ⊗ A of the previous paragraph; this algebra may be regarded as a k-algebra as well as a K-algebra. A k-algebra that is a field is also called a k-field. 1.1.2 Some Important Algebras Endomorphism algebras The archetypal algebra from the viewpoint of representation theory is the n × nmatrix algebra, Matn (k), with unit map sending k to the scalar matrices, or more generally, the algebra Endk (V ) of all k-linear endomorphisms of a vector space V ∈ Vectk . Multiplication in Endk (V ) is given by composition of endomorphisms and the unit map sends each λ ∈ k to the scalar transformation λ Id of V . If V is finitedimensional, say dimk V = n, then any choice of k-basis for V gives rise to a k∼ ∼ linear isomorphism V −→ k⊕n and to an isomorphism of k-algebras Endk (V ) −→ Matn (k). The matrix algebra Matn (k) and the endomorphism algebra Endk (V ) of a finitedimensional vector space V are examples of finite-dimensional algebras, that is, algebras that are finite-dimensional over the base field k.

6

1 Representations of Algebras

Free and tensor algebras We will also often have occasion to consider the free k-algebra that is generated by a given set X; this algebra will be denoted by khXi. One can think of khXi as a noncommutative polynomial algebra over k with the elements of X as noncommuting variables. Assuming X to be indexed, say X = {xi }i∈I , a k-basis of khXi is given by the collection of all finite products, also called monomials or words in the alphabet X, xi1 xi2 . . . xik where (i1 , i2 , . . . , ik ) is a finite (possibly empty) sequence of indices from I, with the empty word being the identity element 1 ∈ khXi. The order of the symbols xij in these products does matter. Multiplication in khXi is defined by concatenation of words. Formally, khXi can be constructed as the tensor algebra T(kX), where kX is the k-vector space of all formal k-linear combinations of the elements of X (Example A.5). Here, the tensor algebra of any vector space V ∈ Vectk is defined as the direct sum M ⊗k def TV = V k≥0

where V ⊗k is the k th tensor power of V as in (B.9); so (B.8) gives dimk V ⊗k = (dimk V )k The unit map of TV is given by the canonical embedding k = V ⊗0 ,→ TV , and multiplication in TV comes from the associativity isomorphisms (B.10) for tensor powers: (v1 ⊗ v2 ⊗ · · · ⊗ vk )(v10 ⊗ v20 ⊗ · · · ⊗ vl0 ) = v1 ⊗ · · · ⊗ vk ⊗ v10 ⊗ · · · ⊗ vl0 for vi , vj0 ∈ V . This multiplication is distributively extended to define products of arbitrary elements of TV . In this way, TV becomes a k-algebra. Note that the subspace V = V ⊗1 ⊆ TV generates the algebra TV in the sense that the only subalgebra of TV containing V is TV itself. Equivalently, every element of TV is a k-linear combination of finite products with factors from V . In fact, any generating set of the vector space V will serve as a set of generators for the algebra TV . The importance of tensor algebras stems from their functorial properties, which we shall now endeavor to explain in some detail. First, the tensor algebra construction gives a functor T : Vectk −→ Algk from the category of k-vector spaces to the category of k-algebras. To any given k-vector space V we associate the k-algebra TV ; this defines the functor T on the objects of the categories under consideration. As for morphisms, let f ∈

1.1 Algebras

7

Homk (V, W ) be a homomorphism of k-vector spaces. Then we have morphisms f ⊗k ∈ Homk (V ⊗k , W ⊗k ) for each k as in §B.1.3: f ⊗k (v1 ⊗ · · · ⊗ vk ) = f (v1 ) ⊗ · · · ⊗ f (vk ) The k-linear map def

Tf =

M

f ⊗k : TV =

k

M

V ⊗k −→

k

M

W ⊗k = TW

k

is easily seen to be a k-algebra map and it is equally straightforward to check that T satisfies all requirements of a functor. The property of the tensor algebra that is expressed in the following proposition is sometimes referred to as the universal property of the tensor algebra; it actually characterizes the tensor algebra and determines it up to isomorphism (Exercise 1.1). Proposition 1.1. Given a k-vector space V and a k-algebra A, there is a natural bijection of sets HomAlg (TV, A) ∼ Homk (V, A ) Vectk

∈

∈

k

f

f V

Here, f V denotes the restriction of f to V = V ⊗1 ⊆ TV . The notation A Vect k indicates that the algebra A is viewed just as a k-vector space, with all other algebra ∼ structure being ignored. We also use the symbol −→ for an an isomorphism in the category Sets, that is, a bijection. The bijection in Proposition 1.1 behaves well with respect to varying the input data, V and A — this is what “naturality” of the bijection is meant to convey. Technically, the functor T : Vectk → Algk and the forgetful functor ? Vect : Algk → Vectk are a pair of adjoint functors. The reader wishing to k see the specifics spelled out is referred to Section A.4 in the appendix on categories. Proof of Proposition 1.1. The map in the proposition is injective, because V gener ates the algebra TV . For surjectivity, let φ ∈ Homk (V, A Vect ) be given. Then the k

map V k → A, (v1 , v2 , . . . , vk ) 7→ φ(v1 )φ(v2 ) · · · φ(vk ) is k-multilinear, and hence it gives rise to a unique k-linear map φk : V ⊗k → A, v1 ⊗· · ·⊗vk 7→ φ(v1 ) · · · φ(vk ) by (B.12). The maps φk yield a unique k-linear map f : TV → A such that f V ⊗k = φk for all k. In particular, f V = φ as needed, and it is also immediate that f is in fact a k-algebra map. This establishes surjectivity. t u Tensor algebras are examples of graded algebras, that is, algebras that are equipped with a meaningful notion of “degree” for their nonzero elements. LIn detail, a k-algebra A is said to be graded or, more precisely, Z+ -graded if A = k∈Z+ Ak 0

0

for k-subspaces Ak ⊆ A such that Ak Ak ⊆ Ak+k for all k, k 0 ∈ Z+ . The nonzero

8

1 Representations of Algebras

elements of Ak are called homogeneous of degree k. 1 An algebra map f : A → B between graded algebras A and B is called a homomorphism of graded algebras if f respects the gradings in the sense that f (Ak ) ⊆ B k for all degrees k. All this applies to the tensor algebra TV , with V ⊗k being the component of degree k. The algebra maps Tf : TV → TW constructed above are in fact homomorphisms of graded algebras. Numerous algebras that we shall encounter below carry a natural grading; see Exercise 1.11 for more background on gradings. Returning to the case where V = kX is the vector space with basis X = {xi }i∈I , the k th tensor power (kX)⊗k has a basis given by the tensors xi1 ⊗ xi2 ⊗ · · · ⊗ xik

∈

∈

for all sequences of indices (i1 , i2 , . . . , ik ) of length k. Sending the above k-tensor to the corresponding word xi1 xi2 . . . xik , we obtain an isomorphism of T(kX) with the free algebra khXi described earlier. The grading of T(kX) by the tensor powers (kX)⊗k makes khXi a graded algebra as well: the homogeneous component of degree k is the k-subspace of khXi that is spanned by the words of length k. This grading is often referred to as the grading by “total in degree”. Proposition 1.1 conjunction with the (natural) bijection Homk (kX, A Vect ) ∼ = HomSets (X, A Sets ) k from (A.4) gives a natural bijection HomAlgk (khXi, A) ∼ HomSets (X, A Sets ) (1.4) f f X

for any k-algebra A. Thus, an algebra map f : khXi → A is determined by the values f (x) ∈ A for the generators x ∈ X, and these values can be freely assigned in order to define f . Often the set X will be finite, say X = {x1 , x2 , . . . , xn }, in which case we will also write khx1 , x2 , . . . , xn i for khXi. Then (1.4) becomes ∼

∈

An

∈

HomAlgk (khx1 , x2 , . . . , xn i, A) f


f (xi ) i

(1.5)

Algebras that have a finite set of generators are called affine. They are exactly the homomorphic images of free algebras khXi generated by a finite set X or, equivalently, the homomorphic images of tensor algebras TV with V finite-dimensional.

Polynomial and symmetric algebras Our next example is the familiar commutative polynomial algebra k[x1 , x2 , . . . , xn ], with unit map sending k to the constant polynomials. Formally k[x1 , x2 , . . . , xn ] can 1

k

th

It will be clear from the context if A denotes the k homogeneous component or the k-fold cartesian product of A.

1.1 Algebras

9

be defined by def

k[x1 , x2 , . . . , xn ] = khx1 , x2 , . . . , xn i/ xi xj − xj xi | 1 ≤ i < j ≤ n




where ( . . . ) denotes the ideal that is generated by the indicated elements. Since these elements are all homogeneous (of degree 2), the total degree grading of the free algebra khx1 , x2 , . . . , xn i passes down to a grading of k[x1 , x2 , . . . , xn ] (Exercise 1.11); the grading thus obtained is the usual total degree grading of the polynomial algebra. The universal property (1.5) of the free algebra yields a corresponding universal property for k[x1 , x2 , . . . , xn ]. Indeed, for any k-algebra A, the set HomAlgk (k[x1 , x2 , . . . , xn ], A) can be identified with the set of all algebra maps f : khx1 , x2 , . . . , xn i → A such that f (xi xj − xj xi ) = 0 for all i, j or, equivalently, f (xi )f (xj ) = f (xj )f
(xi ). Thus, for any k-algebra A, sending an algebra map f to the n-tuple f (xi ) yields a natural bijection of sets n o ∼ HomAlgk (k[x1 , x2 , . . . , xn ], A) −→ (ai ) ∈ An | ai aj = aj ai ∀ i, j (1.6) Letting CommAlgk denote the full subcategory of Algk consisting of all commutative k-algebras, then this becomes a natural bijection, for any A ∈ CommAlgk , ∼

An (1.7)

∈

∈

HomCommAlgk (k[x1 , x2 , . . . , xn ], A) f

f (xi )


i

which is analogous to (1.5). Therefore, k[x1 , x2 , . . . , xn ] is also called the free commutative k-algebra generated by the xi s. As for the free algebra, there is a more general basis-free version of the polynomial algebra: the symmetric algebra of a k-vector space V ; it is defined by def

Sym V = (TV )/I

with I = I(V ) = v ⊗ v 0 − v 0 ⊗ v | v, v 0 ∈ V




Since I is generated by homogeneous elements of TV , it follows that I = L the ideal ⊗k , thereby making Sym V a graded algebra (Exercise 1.11): kI ∩V M M ⊗k Sym V = Symk V = V /I ∩ V ⊗k k≥0

k≥0

Moreover, I ∩ V = 0 — so we may again view V ⊆ Sym V — and we will write v1 v2 · · · vk ∈ Symk V for the image of v1 ⊗ · · · ⊗ vk ∈ V ⊗k . The foregoing yields a functor Sym : Vectk −→ CommAlgk Indeed, Sym V is clearly a commutative k-algebra for every V ∈ Vectk . As for morphisms, note that if f ∈ Homk (V, W ) is a homomorphism of vector spaces, then

10

1 Representations of Algebras

∈

∈

the image of a typical generator v ⊗ v 0 − v 0 ⊗ v ∈ I(V ) under the algebra map Tf ∈ HomAlgk (TV, TW ) is the element f (v) ⊗ f (v 0 ) − f (v 0 ) ⊗ f (v) ∈ I(W ). Thus Tf maps I(V ) to I(W ), and hence Tf passes down to an algebra map Symf : Sym V → Sym W . This is in fact a homomorphism of graded algebras: Symk f = Symf Symk V : Symk V Symk W v 1 v2 · · · vk

f (v1 )f (v2 ) · · · f (vk )

For any commutative k-algebra A, there is a natural bijection HomCommAlgk (Sym V, A) ∼ Homk (V, A Vect ) ∈

∈

k

(1.8)

f

f

V

This follows from Proposition 1.1 exactly as (1.7) was derived from (1.5) earlier. As in Proposition 1.1, “naturality” of the bijection (1.8) can be made precise by stating that the functor Sym : Vectk → CommAlgk is left adjoint to the forgetful functor ? Vect : CommAlgk → Vectk . When V = kX for a set X, then we also have natural k )∼ bijections Homk (kX, A = HomSets (X, A ) by (A.4) and so (1.8) becomes Sets

Vectk

a natural bijection, for any A ∈ CommAlgk ,

∈

∈

HomCommAlgk (Sym kX, A) ∼ HomSets (X, A Sets ) (1.9)

f

f

X

n If X = {x1 , x2 , . . . , xn }, then HomSets (X, A Sets ) ∼ = A and, comparing the above bijection with (1.7), it follows that (Exercise 1.1) Sym kX ∼ = k[x1 , x2 , . . . , xn ] As is well-known (see also Exercise 1.12), a k-basis of the homogeneous component of degree k of the polynomial algebra is given by the so-called “standard monomials” of degree k, X k k x11 x22 . . . xnkn with ki ∈ Z+ and ki = k (1.10) i

Identifying each such monomial with a pattern ∗ · · ∗} | |∗ ·{z · · ∗} | ∗ · · · | ∗| ·{z · · ∗} | ·{z k1

k2

kn

consisting of k stars and n − 1 vertical bars, one sees that   k+n−1 k dimk Sym V = (n = dimk V ) n−1

1.1 Algebras

11

Exterior algebras The exterior algebra ΛV of a k-vector space V is defined by def

with J = J(V ) = v ⊗ v | v ∈ V

ΛV = (TV )/J




L Exactly as for the ideal I of Sym V , one sees that J = k J ∩ V ⊗k and J ∩ V = 0. Thus, we may again view V ⊆ ΛV and ΛV is a graded algebra: M k M ⊗k ΛV = Λ V = V /J ∩ V ⊗k k≥0

k≥0

Writing the canonical map V ⊗k  Λk V = V ⊗k /J ∩ V ⊗k as v1 ⊗ v2 ⊗ · · · ⊗ vk 7→ v1 ∧ v2 ∧ · · · ∧ vk , multiplication in ΛV becomes (v1 ∧ · · · ∧ vk )(v10 ∧ · · · ∧ vl0 ) = v1 ∧ · · · ∧ vk ∧ v10 ∧ · · · ∧ vl0 Once again, as for Sym, one obtains a functor Λ : Vectk −→ Algk and the algebra map Λf : ΛV → ΛW for each f ∈ Homk (V, W ) is a homomorphism of graded algebras: Λk W ∈

Λk V ∈

Λk f = Λf Λk V :

v1 ∧ v2 ∧ · · · ∧ vk

f (v1 ) ∧ f (v2 ) ∧ · · · ∧ f (vk )

The defining relations of the exterior algebra state that v ∧ v = 0 for all v ∈ V ; in words, the exterior product is alternating on elements V . This implies the rule v ∧ w = − w ∧ v for all v, w ∈ V — to see this, expand (v + w) ∧ (v + w) = 0 using v ∧ v = w ∧ w = 0. Inductively, it follows that ab = (−1)kl ba for all a ∈ Λk V and b ∈ Λl V . Using | . | to indicate degrees of homogeneous elements, the latter relation can also be stated as follows: ab = (−1)|a||b| ba

(1.11)

The property expressed by (1.11) is referred to as anticommutativity or gradedcommutativity of the exterior algebra. It follows that, for any given collection of elements v1 , v2 , . . . , vn ∈ V and any permutation s of the indices {1, 2, . . . , n}, vs(1) ∧ vs(2) ∧ · · · ∧ vs(k) = sgn(s) v1 ∧ v2 ∧ · · · ∧ vk

(1.12)

where sgn(s) denotes the sign of the permutation s. Indeed, (1.12) is clear from anticommutativity in case s is a transposition interchanging two adjacent indices; the

12

1 Representations of Algebras

general case is a consequence of the standard fact that these transpositions generate the symmetric group Sn . The foregoing implies that if V has basis {xi }, then the elements xi1 ∧ xi2 ∧ · · · ∧ xik with i1 < i2 < · · · < ik (1.13) generate the k-vector space Λk V . These elements do in fact form a basis of Λk V ; see Exercise 1.12. Therefore, if V is finite-dimensional, then so is ΛV and   dimk V dimk Λk V = k The Weyl algebra The algebra def

A1 (k) = khx, yi/(yx − xy − 1)

(1.14)

is called the first Weyl algebra over k. By a slight abuse of notation, the images of x and y in A1 (k) will also be denoted by x and y; so yx = xy + 1 holds in A1 (k). This relation allows us to write each finite product in A1 (k) with factors x or y as a k-linear combination of ordered products of the form xi y j (i, j ∈ Z+ ). These “standard monomials” therefore generate A1 (k) as k-vector space. One can show that they are in fact linearly independent (Exercise 1.14; see also Examples 1.8 and D.3), and hence the standard monomials form a k-basis of A1 (k). If f : A1 (k) → A is any k-algebra map and a = f (x), b = f (y), then we must have ba − ab − 1 = 0 in A. However, this relation is the only restriction, because it guarantees that the homomorphism khx, yi → A that corresponds to the pair (a, b) ∈ A2 in (1.5) does in fact factor through A1 (k). Thus, we have a bijection, natural in A ∈ Algk , n o 2 HomAlgk (A1 (k), A) ∼ (1.15) = (a, b) ∈ A | ba − ab − 1 = 0 1.1.3 Modules As we have mentioned before, the reader is expected to have had some exposure to modules over rings. In order to pave the way for the dual concept of a “comodule”, to be introduced later in §9.2.1, we now review the basic definitions concerning modules over k-algebras. in the diagrammatic style of §1.1.1, again working in the category Vectk . We will also briefly discuss some issues related to switching sides. Left modules Let A = (A, m, u) be a k-algebra. A left module over A, by definition, is an abelian group (V, +) that is equipped with left action of A, that is, a biadditive map A×V → V , (a, v) 7→ a.v, satisfying the conditions a.(b.v) = (ab).v

and

1A .v = v

1.1 Algebras

13

for all a, b ∈ A and v ∈ V . Putting λv := u(λ).v for λ ∈ k, the group V becomes a k-vector space. The action map is easily seen to be k-bilinear, and hence it corresponds to a k-linear map A ⊗ V → V ; see (B.11). Thus, a left A-module may equivalently be defined as a k-vector space V together with a linear map µ = µV : A ⊗ V → V such that the following two diagrams commute: A⊗A⊗V

m ⊗ IdV

A⊗V and

µ

IdA ⊗µ

A⊗V

k⊗V

u ⊗ IdV

A⊗V

∼

µ

(1.16)

µ

V

V

We will generally suppress µ, using instead simple juxtaposition, µ(a ⊗ v) = av, or the slightly more emphatic µ(a ⊗ v) = a.v as above. Given left A-modules V and W , a homomorphism from V to W is the same as a k-linear map f : V → W such that the following diagram commutes A⊗V

IdA ⊗f

µV

A⊗W µW

(1.17)

f

V

W

Again, this is equivalent to the usual condition. As in Appendices A and B, the set of all A-module homomorphisms f : V → W will be denoted by HomA (V, W ) and the resulting category of left A-modules, a subcategory of Vectk , by A Mod

Note that EndA (V ) = HomA (V, V ) is always a k-subalgebra of Endk (V ). We refrain from reminding the reader in tedious detail of the fundamental module theoretic notions such as submodule, factor module, . . . and we shall also assume familiarity with the isomorphism theorems and other standard facts. We will however remark that, by virtue of the bijection Homk (A ⊗ V, V ) ∼ = Homk (A, Endk (V )) given by Hom-⊗ adjunction (B.14), a left module action µ : A⊗V → V corresponds to an algebra map ρ : A → Endk (V ) In detail, for a given ρ, we may define an action µ by µ(a ⊗ v) := ρ(a)(v) for a ∈ A and v ∈ V . Conversely, from a given action µ, we obtain ρ by defining
ρ(a) := v 7→ µ(a ⊗ v) . Changing sides: opposite algebras Naturally, the category ModA

14

1 Representations of Algebras

of all right modules over a given algebra A (as in Appendices A and B) can be also described by diagrams in Vectk as in the previous paragraph. However, it turns out that right A-modules are essentially the same as left modules over a related algebra, the so-called opposite algebra Aop of A. As a k-vector space, Aop is identical to A but Aop is equipped with a new multiplication ∗ given by a ∗ b = ba for a, b ∈ A. ∼ Alternatively, we may realize Aop as a vector space isomorphic to A via . op : A −→ op op op op op op ∼ A and with multiplication given by a b = (ba) . Clearly, A = A. Now suppose that V is a right A-module with right action µ : V ⊗ A → V . Then we obtain a left Aop -module structure on V by defining µop : Aop ⊗ V → V , µop (aop ⊗ v) = µ(v ⊗ a). Similarly, any left A-module action µ : A ⊗ V → V gives rise to a right Aop -action via µop : V ⊗ Aop → V , µop (v ⊗ aop ) = µ(a ⊗ v). Left Aop -modules become right modules over Aop op ∼ = A in this way. Therefore, we obtain an equivalence of categories (§A.3.3) ModA ≡ Aop Mod Alternatively, in terms of algebra maps, it is straightforward to check as above that a right A-module action V ⊗ A → V correponds to an algebra map A → Endk (V )op . Such a map in turn clearly corresponds to an algebra map Aop → Endk (V )op op ∼ = Endk (V ), and hence to a left Aop -module action on V . Bimodules: tensor products of algebras We will almost exclusively work in the context of left modules, but occasionally we shall also encounter modules that arise naturally as right modules or even as bimodules (§B.1.2). If A and B are k-algebras, then an (A, B)-bimodule is the same as a k-vector space V that is both a left A-module and a right B-module, with module actions 0µ : A ⊗ V → V and µ0 : V ⊗ B → V , such that the following diagram commutes: 0

A⊗V ⊗B

µ ⊗ IdB

V ⊗B

0

0

IdA ⊗µ

µ

(1.18)

0

A⊗V

µ

V

As with right modules, it turns out that (A, B)-bimodules are in fact left modules over some algebra, the algebra in question being the tensor product A⊗B op (§1.1.1). Indeed, suppose that V is an (A, B)-bimodule. As we have remarked above, the module actions correspond to algebra maps α : A → Endk (V ) and β : B op → Endk (V ). Condition (1.18) can be expressed by stating that the images of these maps commute elementwise. The “universal property” of the tensor product of algebras (Exercise 1.10), therefore provides us with a unique algebra map A ⊗ B op → Endk (V ), a ⊗ bop 7→ α(a)β(bop ), and this algebra map in turn corresponds to a left A ⊗ B op module structure on V . Defining morphisms between (A, B)-bimodules to be the

1.1 Algebras

15

same as k-linear maps that are left A-module as well as right B-module maps, we once again obtain a category, A ModB In short, we have an equivalence of categories, A ModB

≡

A⊗B

op

Mod

Example 1.2 (The regular bimodule). Any algebra A carries a natural structure as (A, A)-bimodule, with left and right A-module actions given by left and right multiplication, respectively. Commutativity of (1.18) for these actions is equivalent to the associative law of A. The resulting left, right and bimodule structures will be referred to as the regular structures. We will be primarily concerned with the left regular module structure; it will be denoted by Areg ∈ A Mod in order to avoid any confusion with the algebra A. By the foregoing, we may view the regular (A, A)-bimodule A as a left module over the algebra A ⊗ Aop . Example 1.3 (Bimodule structures on Hom-spaces). For given modules V ∈ A Mod and W ∈ B Mod, where A and B are arbitrary k-algebras, the k-vector space Homk (W, V ) becomes an (A, B)-bimodule by defining (a.f.b)(w) := a.f (b.w) for a ∈ A, b ∈ B, f ∈ Homk (W, V ) and w ∈ W . Thus, Homk (W, V ) becomes a left A ⊗ B op -module. If V, W ∈ A Mod, then we may regard V as a left module over the endomorphism algebra EndA (V ) = HomA (V, V ) and similarly for W . The above bimodule action equips HomA (V, W ) with a (EndA (W ), EndA (V ))bimodule structure. For more background on the Hom-functor, see §B.2.1. 1.1.4 Endomorphism Algebras and Matrices This section serves to deploy some technicalities for later use; it may be skipped at a first reading and referred to as the need arises. Throughout, A denotes an arbitrary k-algebra. LnOur first goal is to describe the endomorphism algebra of a finite direct sum Vi . If all these modules are equal to some V ∈ A Mod, i=1 Vi of left A-modules Ln then we will write i=1 Vi = V ⊕n . In general, the direct sum is equipped with module maps that are given by the various embeddings and projections, µi : Vi ,→

n M i=1

Vi

and

πi :

n M

Vi  Vi

i=1

Explicitly, πi (v1 , v2 , . . . , vn ) = vi and µi (v) = (0, . . . , 0, v, 0, . . . , 0), where v is the ith component on the right. Consider the “generalized n × n matrix algebra”

16

1 Representations of Algebras





 HomA (Vj , Vi )

i,j

 HomA (V1 , V1 ) . . . HomA (Vn , V1 )   .. .. =  . . HomA (V1 , Vn ) . . . HomA (Vn , Vn )

The k-vector space structure of this set is “entry-wise”, using the standard k-linear structure on each HomA (Vj , Vi ) ⊆ Homk (Vj , Vi )L and identifying the generalized matrix algebra with the direct sum of vector spaces i,j HomA (Vj , Vi ). Multiplica


P tion comes from composition: fik gkj = k fik ◦ gkj . Note the reversal of indices: fij ∈ HomA (Vj , Vi ). The identity element of the generalized matrix algebra is the diagonal matrix with entries IdVi . Lemma 1.4. (a) For V1 , V2 , . . . , Vn ∈ A Mod , there is a k-algebra isomorphism   Ln EndA ( i=1 Vi ) ∼ HomA (Vj , Vi ) ∈

∈

i,j



f

πi ◦ f ◦ µj

 i,j

(b) Let V ∈ A Mod . Then V ⊕n becomes a left module over Matn (A) via matrix multiplication and there is an isomorphism of k-algebras, EndMatn (A) (V ⊕n )

f

∈

∼

∈

EndA (V )

f ⊕n =

P

i

µi ◦ f ◦ πi

Ln Proof. (a) Let us put V = i=1 Vi and denote the map in (a) by α; it is clearly k-linear. In fact, α is an isomorphism by (B.13). In order to show that α is an algebra map, we note the relations ( X IdVi for i = j µk ◦ πk = IdV and πi ◦ µj = 0 for i 6= j k Using this, we compute   α(f ◦ g) = πi ◦ f ◦ g ◦ µj   P = πi ◦ f ◦ ( k µk ◦ πk ) ◦ g ◦ µj P  = k (πi ◦ f ◦ µk ) ◦ (πk ◦ g ◦ µj ) = α(f )α(g) Similarly α(1) = 1. This shows that α is a k-algebra homomorphism, proving (a). (b) In componentwise notation, the map f ⊕n is given by (vi ) 7→ (f (vi )) and the “matrix multiplication” action of Matn (A) on V ⊕n is given by

1.1 Algebras

17

X
aij .(vj ) = ( aij .vj ) j

It is straightforward to check that f ⊕n ∈ EndMatn (A) (V ⊕n ) and that the map f 7→ f ⊕n in (b) is a k-algebra map. The inverse map is EndMatn (A) (V ⊕n ) −→ EndA (V ) , g 7→ π1 ◦ g ◦ µ1 P For example, in order to check that i µi ◦π1 ◦g ◦µ1 ◦πi = g, note that g commutes with the operators µi ◦πj ∈ Endk (V ⊕n ), because µi ◦πj is given by the action of the elementary matrix ei,j ∈ Matn (A), with 1 in the (i, j)-position and 0s elsewhere. Therefore, X X µi ◦ π1 ◦ g ◦ µ1 ◦ πi = µi ◦ π1 ◦ µ1 ◦ πi ◦ g = IdV ⊕n ◦g = g i

i

t u

This completes the proof of the lemma.

Next, we turn to a generalization of the familiar fact from linear algebra that the ordinary n × n-matrix algebra Matn (k) is the endomorphism algebra of the vector space k⊕n . In place of k⊕n , we will now consider the n-fold direct sum A⊕n reg of ⊕n Areg ∈ A Mod (Example 1.2). Left A-modules isomorphic to some Areg are called finitely generated free (Example A.5); the standard A-module basis is provided by th the elements ei ∈ A⊕n reg with 1 in the i component and 0s elsewhere. Noncommutative algebras require some care with regard to the side of the module action, which is why opposite algebras make another appearance in the following lemma. ∼ Matn (Aop ) via the matrix transpose. Lemma 1.5. (a) Matn (A)op = (b) There is an algebra isomorphism, given by right matrix multiplication,

∈ xij

∼

EndA (A⊕n reg ) ∈

Matn (A)op



P (ai ) 7→ ( i ai xij )

Proof. (a) We will identify opposite algebras with the originals, but with multiplication ∗ . Consider the map . T : Matn (A)op → Matn (Aop ) sending each maT trix to its transpose; this is clearly a k-linear bijection

satisfying 1opn×n = 1n×n . We need to check that, for X = xij , Y = yij ∈ Matn (A) , the equation (X ∗ Y )T = X T Y T holds Matn (Aop ). But the matrix (X ∗ Y )T = (Y X)T has P P (i, j)-entry ` yj` x`i , while the (i, j)-entry of X T Y T equals ` x`i ∗ yj` . By defop inition of the multiplication in A , these two entries are identical. (b) For each x ∈ A, right multiplication gives map rx : Areg → Areg , rx (a) = ax. Since rx ◦ ry = ryx = rx∗y for x, y ∈ A, the assignment x 7→ rx is an algebra map Aop → EndA (Areg ). This map has inverse EndA (Areg ) → Aop , f 7→ f (1). Hence, op EndA (Areg ) ∼ = A as k-algebras and so

18

1 Representations of Algebras

EndA (A⊕n reg )

∼ =

Lemma 1.4(a)

op op Matn (EndA (Areg )) ∼ = Matn (A ) ∼ = Matn (A) part (a)

It is readily checked that this isomorphism is explicitly given as in the lemma.

t u

Exercises 1.1 (Universal properties). It is a standard fact from category theory that any two left adjoint functors of a given functor are naturally isomorphic; see MacLane [120, p. 85]. Verify this fact in the following cases — both correspond to assertions made in this section. (a) The “universal property” stated in Proposition 1.1 determines the tensor algebra TV up to isomorphism. (b) Comparison of (1.7) and (1.9) gives Sym k{x1 , . . . , xn } ∼ = k[x1 , . . . , xn ]. 1.2 (Short exact sequences). (a) Show that, for any commutative diagram in A Mod, f

U

V h

g

W

h is an isomorphism if and only if f is mono, g is epi and V = Im f ⊕ Ker g. f

g

(b) Let 0 → U −→ V −→ W → 0 be a short exact sequence in A Mod (Appendix B). Show that the following conditions are equivalent; if they hold, the given short exact sequence is said to be split: (i) f splits, that is, f 0 ◦ f = IdU for some f 0 ∈ HomA (V, U ); (ii) g splits, that is, g ◦ g 0 = IdW for some g 0 ∈ HomA (W, V ); (iii) S := Im f = Ker g has a complement, that is, V = S ⊕ C for some submodule C ⊆V. 1.3 (Generators of a module). Let V be a module (left, say) over A ∈ Algk . A subset Γ ⊆ V is said to generate V if the only submodule of V containing Γ is V itself or, equivalently, if every element of V can be written as a finite A-linear P combination v∈Γ av v with av ∈ A. Modules that have a finite generating set are called finitely generated; modules that are generated by one element are called cyclic. (a) Let V be finitely generated. Use Zorn’s Lemma to show that every proper submodule U $ V is contained in a maximal proper submodule M , that is, M $ V and M ⊆ M 0 $ V implies M = M 0 . (b) Let 0 → U → V → W → 0 be a short exact sequence in A Mod (Exercise 1.2). Show that if both U and W are finitely generated, then V is finitely generated as well. Conversely, assuming V to be finitely generated, show that W is finitely generated but this need not hold for U . (Give an example to that effect.) (c) Show that the following are equivalent:

1.1 Algebras

19

(i) V has a generating set consisting of n elements; (ii) V is a homomorphic image of the free left A-module A⊕n reg ; (iii) the left Matn (A)-module V ⊕n as in Lemma 1.4(b) is cyclic. 1.4 (Noetherian modules). A left module V over an algebra A is said to be noetherian if V satisfies the Ascending Chain Condition (ACC) on its submodules: given any sequence U1 ⊆ U2 ⊆ U3 ⊆ . . . of submodules of V , there exists some n such that Un = Un+1 = . . . . (a) Show that ACC has the following equivalent reformulations: (i) All submodules of V are finitely generated. (ii) Every nonempty collection of submodules of V has at least one maximal member (Maximum Condition on submodules). In fact, ACC and the Maximum Condition may be formulated for any partially ordered set and they are equivalent in this more general setting (assuming the Axiom of Choice). (b) Let 0 → U → V → W → 0 be a short exact sequence in A Mod (Appendix B). Show that V is noetherian if and only if both U and W are. 1.5 (Noetherian algebras ). The algebra A is called left noetherian if Areg ∈ A Mod is noetherian, that is, A satisfies ACC on left ideals. Right noetherian algebras are defined similarly using right ideals. Algebras that are both right and left noetherian are simply called noetherian. (a) Assuming A to be left noetherian, show that all finitely generated left Amodules are noetherian. (b) Let B be a subalgebra of A such that the k-algebra A is generated by B and an element x such that Bx + B = xB + B. Adapt the proof of the Hilbert Basis Theorem to show that if B is left (or right) noetherian, then A is left (respectively, right) noetherian. 1.6 (Skew polynomial algebras). Let A be a k-algebra. Like the ordinary polynomial algebra over A, a skew polynomial algebra over A is a k-algebra, B, containing A as a subalgebra and an additional element x ∈ B whose powers form a basis of B as P left A-module. Thus, every element of B can be uniquely written as a finite sum i ai xi with ai ∈ A. Contrary to the case of ordinary polynomial algebras, however, we now only insist on the inclusion xA ⊆ Ax + A to hold; so all products xa with a ∈ A can be written in the form xa = σ(a)x + δ(a) with unique σ(a), δ(a) ∈ A. (a) Show that the above rule defines a k-algebra multiplication on B if and only if σ ∈ EndAlgk (A) and δ is a k-linear endomorphism of A satisfying δ(aa0 ) = σ(a)δ(a0 ) + δ(a)a0

(a, a0 ∈ A)

20

1 Representations of Algebras

Maps δ as above are called left σ-derivations of A; if σ = IdA , then one simply speaks of a derivation. The resulting algebra B is commonly denoted by A[x; σ, δ] and one also writes A[x; δ] = A[x; IdA , δ] and A[x; σ] = A[x; σ, 0]. Assuming σ to be bijective, show: (b) If A is a domain (i.e., products of any two nonzero elements of A are nonzero), then A[x; σ, δ] is likewise. (c) If A is left (right) noetherian, then so is A[x; σ, δ]. (Use Exercise 1.5.) 1.7 (Artin-Tate Lemma). Let A be an affine k-algebra and let B be a k-subalgebra such that A is finitely generated as a left B-module, say A = Ba1 + · · · + Bam . P (a) Show that there exists an affine k-subalgebra B 0 ⊆ B such that A = i B 0 ai . (b) Conclude from (a) and the Hilbert Basis Theorem that if B is commutative, then B is affine. 1.8 (Subalgebras as direct summands). Let A be a k-algebra and let B be a ksubalgebra such that A is free as a left B-module. Show that A = B ⊕ C for some left B-submodule C ⊆ A 1.9 (Affine algebras and finitely generated modules). Let A be an affine k-algebra and let M ∈ A Mod be finitely generated. Show that if N is an A-submodule of M such that dimk M/N < ∞, then N is finitely generated. 1.10 (Tensor product of algebras). Let A, B ∈ Algk . Prove: (a) The tensor product A ⊗ B ∈ Algk , as defined in (1.3), has the following universal property: The maps a : A → A ⊗ B, x 7→ x ⊗ 1, and b : B → A ⊗ B, y 7→ 1⊗y, are k-algebra maps such that Im a commutes elementwise with Im b. Moreover, if α : A → C and β : B → C are any k-algebra maps such that Im α commutes elementwise with Im β, then there exists a unique k-algebra map t : A ⊗ B → C satisfying t ◦ a = α and t ◦ b = β. α

A

a

A⊗B B

∃! t

C

b β

In particular, the tensor product gives a bifunctor · ⊗ · : Algk × Algk −→ Algk . (b) Z (A ⊗ B) = Z (A) ⊗ Z (B). (c) C ⊗R H ∼ = Mat2 (C) as C-algebras. (d) A ⊗ k[x1 , . . . , xn ] ∼ = A[x1 , . . . , xn ] as k-algebras. In particular, k[x1 , . . . , xn ] ⊗ k[x1 , . . . , xm ] ∼ = k[x1 , . . . , xn+m ] (e) A ⊗ Matn (k) ∼ = Matn (A) as k-algebras. In particular, Matn (k) ⊗ Matm (k) ∼ = Matnm (k)

1.1 Algebras

21

1.11 (Graded vector spaces, algebras and modules). Let ∆ be any set. A ∆grading of a k-vector space V is given by a direct sum decomposition M k V = V k∈∆

with k-subspaces V k . The nonzero elements of V k are said to be homogeneous of degree k. If V and W are ∆-graded, then a morphisms f : V → W of ∆-graded vector spaces, by definition, is a k-linear map that preserves degrees in the sense that f (V k ) ⊆ W k for all k ∈ ∆. In this way, ∆-graded k-vector spaces form a category, Vect∆ k (a) LetL V be a ∆-graded k-vector space and let U be any k-subspace of V . Show that U = k (U ∩ V k ) if and only if U is generated by homogeneous elements as a k-vector space. In this case, the k-vector space V /U is graded with homogeneous components (V /U )k = V k /U ∩ V k . Now assume that ∆ is a monoid, with binary operation denoted by juxtaposition and with identity element 1, and let V, W ∈ Vect∆ k . Then the tensor product V ⊗ W inherits a grading with homogeneous component of degree k given by M i (V ⊗ W )k = V ⊗ Wj ij=k

A k-algebra A is said to be ∆-graded if the underlying k-vector space of A is ∆graded and multiplication A ⊗ A → A as well as the unit map k → A are morphisms of graded vector spaces. Here, k has the trivial grading: k = k1 . Explicitly, this 0 0 L means that A = k∈∆ Ak for k-subspaces Ak satisfying Ak Ak ⊆ Akk for k, k 0 ∈ ∆ and 1A ∈ A1 . In particular A1 is a k-subalgebra of A. Taking as morphisms the algebra maps that preserve the ∆-grading, we obtain a category Alg∆ k (b) Assuming that k 6= 1 implies kk 0 6= k 0 for all k 0 ∈ ∆, show that 1 ∈ A1 is in fact automatic if multiplication of A is a map of graded L vectorkspaces. (c) Let I be an ideal of A. Show that I = k (I ∩ A ) if and only if I is generated, as an ideal of A, by homogeneous elements. In this case, the algebra A/I is graded with homogeneous components (A/I)k = Ak /I ∩ Ak . Assuming A to be ∆-graded, an A-module V (left, say) is called ∆-graded if if the underlying k-vector space of V is ∆-graded L and the action map A ⊗ V → V is a morphism of graded vector spaces. Thus, V = k∈∆ V k for k-subspaces V k such 0

0

that Ak V k ⊆ V k+k for all k, k 0 ∈ ∆. L (d) Let U be a submodule of V . Show that U = k (U ∩ V k ) if and only if U is generated, as A-module, by homogeneous elements. In this case, the A-module V /U is graded with homogeneous components (V /U )k = V k /U ∩ V k .

22

1 Representations of Algebras

1.12 (Some properties of symmetric and exterior algebras). Let V, W ∈ Vectk . (a) Show that Sym(V ⊕ W ) ∼ = Sym V ⊗ Sym W as graded k-algebras. (Use Exercise 1.10(a) and the universal property (1.8) of the symmetric algebra.) L k (b) A Z-graded algebra A = k∈Z A is called anticommutative or gradedcommutative if A satisfies the condition ab = (−1)|a||b| ba as in (1.11) . If, in addition, a2 = 0 for all homogeneous elements a ∈ A of odd degree, then A is called alternating. (Note that anticommutative algebras are automatically alternating if char k 6= 2.) Show that the exterior algebra ΛV is alternating and that, for any alternating k-algebra A, there is a natural bijection of sets HomAlgZ (ΛV, A) ∼ Homk (V, A1 ) ∈

∈

k

f

f V

where AlgZ k denotes the category of Z-graded algebras as in Exercise 1.11. (c) Let A and B be alternating Z-graded k-algebras. Define A g ⊗ B to be the usual tensor product A ⊗ B as a k-vector space, with the Z-grading of Exercise 1.11, but with multiplication given by the “Koszul sign rule” 0

(a ⊗ b)(a0 ⊗ b0 ) := (−1)|b||a | aa0 ⊗ bb0 instead of (1.3). Show that this makes A g⊗ B an alternating k-algebra. g (d) Conclude from (b) and (c) that Λ(V ⊕W ) ∼ = ΛV ⊗ΛW as graded k-algebras. (e) Deduce from the isomorphisms in (a) and (d) the bases of Sym V and ΛV as stated in (1.10) and (1.13). 1.13 (Central simple algebras). A k-algebra A 6= 0 is called simple if 0 and A are the only ideals of A. Show that this implies that the center Z A is a k-field. The k-algebra A is called central simple if A is simple and Z A = k, viewing the unit map k → A as an embedding. (In the literature, central simple k-algebras are often also understood to be finite-dimensional over k, but we will not assume this.) (a) Show that if A is a central simple k-algebra and B is any k-algebra, then the ideals of the algebra A ⊗ B are exactly the subspaces of the form A ⊗ I, where I is an ideal of B. (b) Show that the endomorphism algebra Endk (V ) of V ∈ Vectk is central simple if and only if V is finite-dimensional. Conclude from (a) and Exercise 1.10(e) that the ideals of the matrix algebra Matn (B) are exactly the various Matn (I) with I an ideal of B. (c) Conclude from (a) and Exercise 1.10(b) that the tensor product of any two central simple algebras is again central simple. 1.14 (Weyl algebras). Let A1 (k) denote the Weyl algebra, with standard algebra generators x and y and defining relation yx = xy + 1 as in (1.14).

1.2 Representations

23

(a) Consider the skew polynomial algebra B = A[η; δ] (Exercise 1.6) with A = d the “differentiation” derivation k[ξ] the ordinary polynomial algebra and with δ = dξ of A. Show that ηξ = ξη + 1 holds in B. Conclude from (1.15) that there is a unique algebra map f : A1 (k) → B with f (x) = ξ and f (y) = η. Conclude further that f is an isomorphism and that the standard monomials xi y j form a k-basis of A1 (k). Finally, conclude from Exercise 1.6 that A1 (k) is a noetherian domain. (b) Assuming char k = 0, show that A1 (k) is central simple in the sense of Exercise 1.13. Conclude from Exercise 1.13 that the algebra An (k) := A1 (k)⊗n is central simple for every positive integer n; this algebra is called the nth Weyl algebra. (c) Now let char k = p > 0. Show that Z := Z (A1 (k)) = k[xp , y p ] is a 2 ⊕p polynomial algebra in two variables over k and that A1 (k) ∼ as Z -module: =Z the standard monomials xi y j with 0 ≤ i, j < p form a Z -basis. 1.15 (Quantum plane and quantum torus). Fix a scalar q ∈ k× and consider the following algebra, called the quantum plane, def

Oq (k2 ) = khx, yi/(xy − q yx) As in the case of the Weyl algebra A1 (k), denote the images of x and y in Oq (k2 ) by x and y as well; so xy = q yx holds in Oq (k2 ). (a) Adapt the method of Exercise 1.14(a) to show that the quantum plane can be realized as the skew polynomial algebra Oq (k2 ) ∼ = k[x][y; σ], where k[x] is the ordinary polynomial algebra and σ ∈ AutAlgk (k[x]) is given by σ(x) = q −1 x. Conclude from Exercise 1.6 that Oq (k2 ) is a noetherian domain. (b) Observe that σ extends to an automorphism of the Laurent polynomial algebra k[x±1 ]. Using this fact, show that there is a tower of k-algebras k[x][y; σ] ⊆ k[x±1 ][y; σ] ⊆ k[x±1 ][y ±1 ; σ] where the last algebra is a skew Laurent polynomial algebra – it is defined exactly like the the skew polynomial algebra k[x±1 ][y; σ] (Exercise 1.6) except that negative powers of the variable y are permitted. The algebra k[x±1 ][y ±1 ; σ] is called a quantum torus and often denoted by Oq ((k× )2 ). (c) Show, that if the parameter q is not a root of unity, then Oq ((k× )2 ) is a central simple k-algebra. Conclude that every nonzero ideal of the quantum plane Oq (k2 ) contains some standard monomial xi y j . (d) If q is a root of unity of order n, then show that Z := Z (Oq ((k× )2 )) is a Laurent polynomial algebra in the two variables x±n , y ±n and the standard monomials xi y j with 0 ≤ i, j < n form a basis of Oq ((k× )2 ) as module over Z .

1.2 Representations By definition, a representation of a k-algebra A is an algebra homomorphism ρ : A → Endk (V ), where V is some k-vector space. If dimk V = n is finite, then

24

1 Representations of Algebras

the representation is called a finite-dimensional representation and n is referred to as its dimension or the degree. We will usually denote the operator ρ(a) by aV ; so

∈

Endk (V )

∈

ρ: A a

aV

(1.19)

The map ρ is often de-emphasized and the space V itself is referred to as a representation of A. For example, we will write Ker V instead of Ker ρ: def

Ker V = {a ∈ A | aV = 0} Representations with kernel 0 are called faithful. The image ρ(A) of a representation (1.19) will be written as AV ; so A/ Ker V ∼ = AV ⊆ Endk (V ) Throughout the remainder of this section, A will denote an arbitrary k-algebra unless explicitly specified otherwise. 1.2.1 The Category Rep A and First Examples As we have explained in §1.1.3, representations of A are essentially the same as left A-modules: every representation A → Endk (V ) gives rise to a left A-module action A ⊗ V → V and conversely. This connection enables us to transfer familiar notions from the theory of modules into the context of representations. Thus, we may speak of subrepresentations, quotients and direct sums of representations and also of homomorphisms, isomorphisms, . . . of representations by simply using the corresponding definitions for modules. Formally, the representations of an algebra A form a category, Rep A, that is equivalent to the category of left A-modules: Rep A

≡

A Mod

For example, a homomorphism from a representation ρ : A → Endk (V ) to a representation ρ0 : A → Endk (V 0 ) is given by an A-module homomorphism f : V → V 0 . Explicitly, f is a k-linear map satisfying the condition ρ0 (a) ◦ f = f ◦ ρ(a)

(a ∈ A)

(1.20)

0

This is sometimes stated as “f intertwines ρ and ρ .” An isomorphism from ρ to ρ0 ∼ is given by a k-vector space isomorphism f : V −→ V 0 satisfying the intertwining condition (1.20), which amounts to commutativity of the following diagram in Algk : Endk (V )

∼ f ◦ · ◦f

−1

Endk (V 0 ) 0

ρ

ρ

A

(1.21)

1.2 Representations

25

Here, f ◦ · ◦f −1 = f∗ ◦(f −1 )∗ in the notation of §B.2.1. Isomorphic representations are also called equivalent, and the symbol ∼ = is used for equivalence or isomorphism of representations. In the following, we shall freely use module theoretic notation and terminology for representations. For example, ρ(a)(v) = aV (v) will usually be written as a.v or simply av. Example 1.6 (Regular representations). The representation of A that corresponds to the module, Areg ∈ A Mod as defined in Example 1.2, is called the regular representation of A; it is given by the algebra map ρreg : A → Endk (A) with ρreg (a) = aA := (b 7→ ab)

(a, b ∈ A)

As in Example 1.2, we may also consider the right regular A-module as well as the regular (A, A)-bimodule; these correspond to representations Aop → Endk (A) and A ⊗ Aop → Endk (A), respectively. Example 1.7 (The polynomial algebra). Let us consider the case where A = k[t] is the ordinary polynomial algebra. By (1.6) representations ρ : k[t] → Endk (V ) are in bijection with linear operators τ ∈ Endk (V ) via ρ(t) = τ . For a fixed positive integer n, we may describe the equivalence classes of n-dimensional representations of k[t] as follows. Given any such representation, V , we may choose an isomorphism ∼ V −→ k⊕n and replace τ by a matrix T ∈ Matn (k). By (1.21) the representations of k[t] that are given by T, T 0 ∈ Matn (k) are equivalent if and only if the matrices T and T 0 are conjugate to each other. Thus, we have a bijection of sets equivalence classes of { n-dimensional } representations of k[t]

∼

GLn (k)\ Matn (k)

(1.22)

where GLn (k) is the group of invertible n×n-matrices over k and GLn (k)\ Matn (k) denotes the set of orbits for the conjugation action GLn (k) Matn (k) that is given by g.T = gT g −1 . From linear algebra, we further know that a full representative set of the GLn (k)-orbits in Matn (k) is given by the matrices in rational canonical form in Matn (k) or the Jordan canonical form, up to a permutation of the Jordan blocks, over some algebraic closure of k; see [55, Chapter 12, Theorems 16 and 23]. Example 1.8 (The standard representation of the Weyl algebra). In view of (1.15), representations V of the Weyl algebra A1 (k) = khx, yi/(yx − xy − 1) correspond to pairs (a, b) ∈ Endk (V )2 satisfying the relation ba = ab + IdV . As in Example 1.7, it follows from (1.21) that two such pairs (a, b), (a0 , b0 ) ∈ Endk (V )2 yield equivalent representations A1 (k) → Endk (V ) if and only if they belong to the same orbit of × the diagonal conjugation action GL(V ) Endk (V )2 , where GL(V ) = Endk (V ) is the group of invertible linear transformations of V : (a0 , b0 ) = g.(a, b) = (gag −1 , gbg −1 ) for some g ∈ GL(V ). The standard representation of A1 (k) is obtained by taking V = k[t] and the two k-linear endomorphisms of the polynomial algebra k[t] that

26

1 Representations of Algebras

d are given by multiplication with the variable t and by formal differentiation, dt . Ded d noting the former by just t, the product rule gives dt t = t dt + Idk[t] as required. d Thus, we obtain a representation A1 (k) → Endk (k[t]) with x 7→ t and y 7→ dt . It is

elementary to see that the operators ti

j

d j dt

∈ Endk (k[t]) (i, j ∈ Z+ ) are k-linearly in-

dependent if char k = 0 (Exercise 1.22). It follows that the standard monomials xi y j form a k-basis of A1 (k), at least when char k = 0; for general k, see Exercise 1.14 or Example D.3. 1.2.2 Changing the Algebra or the Base Field In studying the representations of a given k-algebra A, it is often useful to extend the base field k — things tend to become simpler over an algebraically closed or at least sufficiently large field — or to take advantage of any available information concerning the representations of certain related algebras such as subalgebras or homomorphic images of A. Here we describe some standard ways to go about doing this. This material is somewhat technical; it can be skipped or only briefly skimmed at a first reading and referred to later as needed. Pulling back: restriction Suppose we are given a k-algebra map φ : A → B. Then any representation ρ : B → Endk (V ), b 7→ bV , gives rise to a representation φ∗ (ρ) := ρ ◦ φ : A → Endk (V ); so aV = φ(a)V for a ∈ A. We will refer to this process as pulling back the representation ρ along φ; the resulting representation φ∗ (ρ) of A is also called the the restriction of ρ from B to A. The “restriction” terminology is especially intuitive in case A is a subalgebra of B and φ is the embedding, or if φ is at least a monomorphism, but it is also used for general φ. If φ is an epimorphism, then φ∗ (ρ) is sometimes referred to as the inflation of ρ along φ. In keeping with the general tendency to emphasize V over the map ρ, the pull-back φ∗ (ρ) is often denoted by φ∗ V . When φ is understood, we will also write V ↓A or ResB AV The process of restricting representations along a given algebra map clearly is functorial: any morphism ρ → ρ0 in Rep B gives rise to a homomorphism φ∗ (ρ) → φ∗ (ρ0 ) in Rep A, because the intertwining condition (1.20) for ρ and ρ0 is clearly inherited by φ∗ (ρ) and φ∗ (ρ0 ). In this way, we obtain the restriction functor ResB A : Rep B → Rep A Pushing forward: induction and coinduction In the other direction, we may also “push forward” representations along an algebra map φ : A → B. In fact, there are two principal ways to do this. First, for any V in Rep A, the induced representation of B is defined by

1.2 Representations

27

def

IndB A V = B ⊗A V On the right, B carries the (B, A)-bimodule structure that comes from the regular (B, B)-bimodule structure via b.b0.a := bb0 φ(a). As in §B.1.2, this allows us to form the tensor product B ⊗A V and equip it with the left B-module action b.(b0 ⊗ v) := bb0 ⊗ v B making IndB A V a representation of B. Alternative notations for IndA V include φ∗ V B 0 and V ↑ . Again, this construction is functorial: if f : V → V is a morphism in B B 0 Rep A, then IndB A f := IdB ⊗f : IndA V → IndA V is a morphism in Rep B. All this behaves well with respect to composition and identity morphisms; so induction gives a functor IndB A : Rep A → Rep B

Similarly, we may use the (A, B)-bimodule structure of B that is given by a.b0.b := φ(a)b0 b to form HomA (B, V ) and view it as left B-module as in §B.2.1: (b.f )(b0 ) = f (b0 b) The resulting representation of B is called the coinduced representation: def

CoindB A V = HomA (B, V ) B If f : V → V 0 is a morphism in Rep A, then CoindB A f := f∗ : CoindA V → B 0 CoindA V , g 7→ f ◦ g, is a morphism in Rep B. The reader will have no difficulty to ascertain that this gives a functor

CoindB A : Rep A → Rep B Adjointness relations B B It turns out that the functors IndB A and CoindA are left and right adjoint to ResA , respectively, in the sense of Section A.4. These abstract relations have very useful consequences; see Exercises 1.21 and 1.26 for a first example. The isomorphism in (a) and various consequences thereof are often referred to as Frobenius reciprocity.

Proposition 1.9. Let φ : A → B be a map of k-algebras. Then, for any V ∈ Rep A and W ∈ Rep B, there are natural isomorphisms of k-vector spaces B ∼ (a) HomB (IndB A V, W ) = HomA (V, ResA W ) B ∼ (b) HomB (W, CoindB A V ) = HomA (ResA W, V )

28

1 Representations of Algebras

Proof. Both parts follow from the more general Hom-⊗ isomorphism (B.15). For (a), we use the (B, A)-bimodule structure of B that was explained above to form HomB (B, W ) and equip it with the left A-action (a.f )(b) := f (bφ(a)). In particular, (a.f )(1) = f (φ(a)) = φ(a).f (1); so the map f 7→ f (1) is an isomorphism ∼ HomB (B, W ) −→ ResB A W in Rep A. Therefore, HomB (IndB A V, W ) = HomB (B ⊗A V, W ) ∼ = HomA (V, HomB (B, W )) (B.15)

B ∼ = HomA (V, ResA W )

Tracking a homomorphism f ∈ HomA (V, ResB A W ) through the above isomorphisms, one obtains the map in HomB (IndB V, W ) that is given by b ⊗ v 7→ b.f (v) A for b ∈ B and v ∈ V . Part (b) uses the above (A, B)-bimodule structure of B and the standard Bmodule isomorphism B ⊗B W ∼ = W . This isomorphism restricts to an isomorphism B B ⊗B W ∼ = ResA W in Rep A, giving HomB (W, CoindB A V ) = HomB (W, HomA (B, V )) ∼ = HomA (B ⊗B W, V ) (B.15)

B ∼ = HomA (ResA W, V )

t u Twisting representations For a given V ∈ Rep A, we may use restriction or induction along some α ∈ AutAlgk (A) to obtain a new representation of A, called a twist of V . For restriction, each a ∈ A acts on V via α(a)V . Using induction instead, we have ∼ α∗ V = IndA A V = A ⊗A V = V , with 1 ⊗ v ↔ v, and a.(1 ⊗ v) = a.1 ⊗ v = a ⊗ v = 1.α−1 (a) ⊗ v = 1 ⊗ α−1 (a).v −1 Thus, identifying IndA (a)V . A V with V as above, each a ∈ A acts on V via α Consequently, the difference between twisting by restriction or induction is minor; α we shall generally work with the latter. Putting α V := IndA A V and v := 1 ⊗ v, we α ∼ α have : V −→ V as k-vector spaces and the result of the above calculation can be restated as a.α v = α (α−1 (a).v) or, equivalently α

(a.v) = α(a).α v

(a ∈ A, v ∈ V )

(1.23)

Extending the base field For any field extension K/k and any representation ρ : A → Endk (V ) of A, we may consider the representation of the K-algebra K ⊗ A that is obtained from ρ by

1.2 Representations

29

“extension of scalars”. The resulting representation may be described as the representation IndK⊗A V = (K ⊗ A) ⊗A V ∼ = K ⊗ V that comes from the k-algebra map A A → K ⊗ A, a 7→ 1 ⊗ a. However, we view K ⊗ A as a K-algebra as in §1.1.1, and hence we move from Algk to AlgK in the process. Explicitly, the action of K ⊗ A on K ⊗ V is given by (λ ⊗ a).(λ0 ⊗ v) = λλ0 ⊗ a.v for λ, λ0 ∈ K, a ∈ A and v ∈ V ; equivalently, in terms of algebra homomorphisms, IdK ⊗ρ

K ⊗ Endk (V )

can.

EndK (K ⊗ V ) ∈

K ⊗A

∈

K ⊗ ρ:

λ⊗f

ρreg (λ) ⊗ f

(1.24)

The “canonical” map in (1.24) is a special case of (B.26); this map is always injective, and it is an isomorphism if V is finite-dimensional or the field extension K/k is finite. Example 1.10 (The polynomial algebra). Recall from Example 1.7 that the equivalence classes of n-dimensional representations of k[t] are in bijection with the set of orbits for the conjugation action GLn (k) Matn (k). It is a standard fact from linear algebra (e.g., [55, Chapter 12, Corollary 18]) that if two matrices T, T 0 ∈ Matn (k) belong to the same GLn (K)-orbit for some field extension K/k, then T, T 0 also belong to the same GLn (k)-orbit. In other words, denoting the corresponding repre0 sentations of k[t] by V and V 0 , we know that if K ⊗ V ∼ = K ⊗ V as representations 0 ∼ of K[t], then we must have V = V to start with. For a generalization of this fact, see the Noether-Deuring Theorem (Exercise 1.20). 1.2.3 Irreducible Representations A representation ρ : A → Endk (V ), a 7→ aV , is said to be irreducible if V is an irreducible A-module. Explicitly, this means that V 6= 0 and no k-subspace of V other than 0 and V is stable under all operators aV with a ∈ A; equivalently, it is impossible to find a k-basis of V such that the matrices of all operators aV have block upper triangular form  

∗

 

0

∗

∗

 

Irreducible representations are also called simple and they are often informally referred to as “irreps.” Example 1.11 (Division algebras). Recall that a division k-algebra is a k-algebra D 6= 0 whose nonzero elements are all invertible: D× = D \ {0}. Representations of D are the same as left D-vector spaces, and a representation V is irreducible if and only if dimD V = 1. Thus, up to equivalence, the regular representation of D is the only irreducible representation of D.

30

1 Representations of Algebras

Example 1.12 (Tautological representation of EndD (V )). For any k-vector space V 6= 0, the representation of the algebra Endk (V ) that is given by the identity map Endk (V ) → Endk (V ) is irreducible. For, if u, v ∈ V are given, with u 6= 0, then there exists f ∈ Endk (V ) such that f (u) = v. Therefore, any nonzero subspace of V that is stable under all f ∈ Endk (V ) must contain all v ∈ V . The foregoing applies verbatim to any nonzero representation V of a division k-algebra D: the embedding EndD (V ) ,→ Endk (V ) is an irreducible representation of the algebra EndD (V ). If dimD V < ∞, then this representation is in fact the only irreducible representation of EndD (V ) up to equivalence; this is a consequence of Wedderburn’s Structure Theorem (§1.4.3). Example 1.13 (The standard representation of the Weyl algebra). Recall from Example 1.8 that the standard representation of A1 (k) is the algebra homomorphism d . If A1 (k) → Endk (V ), with V = k[t], that is given by xV = t· and yV = dt char k = 0, then the standard representation is irreducible. To see this, let U ⊆ V be any nonzero subrepresentation of V and let 0 6= f ∈ U be a polynomial of minimal degree among all nonzero polynomials in U . Then f ∈ k× ; for, if deg f > 0, then d d f = y.f ∈ U and dt f has smaller degree than f . Therefore, k ⊆ U and 0 6= dt repeated application of xV gives that all ktn ⊆ U . This shows that U = V , proving irreducibility of the standard representation for char k = 0. There are many more irreducible representations of A1 (k) in characteristic 0; see Block [15]. For irreducible representations of A1 (k) in positive characteristics, see Exercise 1.22. One of the principal, albeit often unachievable, goals of representation theory is to provide, for a given k-algebra A, a good description of the following set: def

Irr A = the set of equivalence classes of irreducible representations of A Of course, Irr A can also be thought of as the set of isomorphism classes of irreducible left A-modules. We will also use Irr A to denote a full set of representatives of the isomorphism classes of irreducible left A-modules and S ∈ Irr A will indicate that S is an irreducible representation of A. To see that Irr A is indeed a set, we observe that every irreducible representation of A is a homomorphic image of the regular representation Areg . To wit: Lemma 1.14. A full representative set for Irr A is furnished by the non-equivalent factors Areg /L, where L is a maximal left ideal L of A. In particular, dimk S ≤ dimk A for all S ∈ Irr A. Proof. If S is an irreducible representation of A, then any 0 6= s ∈ S gives rise to a homomorphism of representations f : Areg → S, a 7→ as. Since Im f is a nonzero subrepresentation of S, it must be equal to S. Thus, f is an epimorphism and S ∼ = Areg /L with L = Ker f ; this is a maximal left ideal of A by irreducibility of S. Conversely, all factors of the form Areg /L, where L is a maximal left ideal of A, are irreducible left A-modules, and hence we may select our equivalence classes

1.2 Representations

31

of irreducible representations of A from the set of these factors. The last assertion of the lemma is now clear. t u Example 1.15 (The polynomial algebra). By Example 1.7, a representation V ∈ Rep k[t] corresponds to an endomorphisms τ = tV ∈ Endk (V ). Lemma 1.14 fur∼ ther tells us that irreducible representations of k[t]
have the form V = k[t]reg /L, where L is a maximal ideal of k[t]; so L = m(t) for some irreducible polynomial m(t) ∈ k[t]. Note that m(t) is both the characteristic and the minimal polynomial of τ = tV . Thus, an irreducible representation of k[t] corresponds to an endomorphism τ ∈ Endk (V ) with V ∈ Vectk finite-dimensional and such that the characteristic and the minimal polynomial of τ coincide and are irreducible. In particular, if k is algebraically closed, then all irreducible representations of k[t] are 1-dimensional. 1.2.4 Composition Series Every finite-dimensional representation V 6= 0 of A can be assembled from irreducible pieces in the following way. To start, pick some irreducible subrepresentation V1 ⊆ V ; any nonzero subrepresentation of minimal dimension will do. If V1 6= V , then we may similarly choose an irreducible subrepresentation of V /V1 , which will have the form V2 /V1 for some subrepresentation V2 ⊆ V . If V2 6= V , then we continue in the same manner. Since V is finite-dimensional, the process must stop after finitely many steps, resulting in a finite chain 0 = V0 ⊂ V1 ⊂ · · · ⊂ Vl = V

(1.25)

of subrepresentations Vi such that all Vi /Vi−1 are irreducible. An analogous construction can sometimes be carried out even when V is not necessarily finite-dimensional (Exercises 1.23, 1.24). Any chain of the form (1.25), with irreducible factors V i = Vi /Vi−1 , is called a composition series of V and the number l is called the length of the series. If the composition series (1.25) is given, then choosing a k-basis of V that contains a basis of each Vi , the matrices of all operators aV (a ∈ A) have block upper triangular form, with (possibly infinite) diagonal blocks coming from the irreducible representations V i :   aV 1       a V2     ..   .     aV l

∗

0

Example 1.16 (The polynomial algebra). Let V ∈ Rep k[t] be finite-dimensional and assume that k is algebraically closed. Then, in view of Example 1.15, fixing a composition series for V amounts to the familiar process of choosing a k-basis of V such that the matrix of the endomorphism tV ∈ Endk (V ) is upper triangular with the eigenvalues of tV along the diagonal.

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1 Representations of Algebras

Example 1.17 (Composition series need not exist). If A is any domain (not necessarily commutative) that is not a division algebra, then the regular representation Areg does not have a composition series; in fact, Areg does not contain any irreducible subrepresentations. To see this, observe that subrepresentations of Areg are the same as left ideals of A. Moreover, if L is any nonzero left ideal of A, then there exists some 0 6= a ∈ L with a ∈ / A× . Then L ⊇ Aa % Aa2 6= 0, showing that L is not irreducible. Representations that admit a composition series are said to be of finite length. The reasoning behind this terminology will be clearer shortly; for now, we just remark that finite-length representations of a division algebra D are the same as finitedimensional left D-vector spaces by Example 1.11. For any algebra, the class of representations of finite length behaves quite well in several respects. Most importantly, all composition series of such a representation are very much alike; this is the content of the celebrated Jordan-Hölder Theorem, which is stated as part (b) of the theorem below. Part (a) shows that the property of having finite length also transfers well in short exact sequences in Rep A, f

g

0 → U −→ V −→ W → 0

(1.26)

Here, “exactness” has the same meaning as in Appendix B: f is injective, g is surjective, and Im f = Ker g. Theorem 1.18. (a) Given a short exact sequence (1.26) in Rep A, the representation V has finite length if and only if both U and W do. (b) Let 0 = V0 ⊂ V1 ⊂ · · · ⊂ Vl = V and 0 = V00 ⊂ V10 ⊂ · · · ⊂ Vl00 = V be two composition series of V ∈ Rep A . Then l = l0 and there exists a permutation s 0 0 of {1, . . . , l} such that Vi /Vi−1 ∼ = Vs(i) /Vs(i)−1 for all i . Proof. (a) First, assume that U and W have finite length and fix composition series 0 = U0 ⊂ U1 ⊂ · · · ⊂ Ur = U and 0 = W0 ⊂ W1 ⊂ · · · ⊂ Ws = W . These series can be spliced together to obtain a composition series for V as follows. Put Xi = f (Ui ) and Yj = g −1 (Wj ). Then Xi /Xi−1 ∼ = Ui /Ui−1 via f and Yj /Yj−1 ∼ = Wj /Wj−1 via g. Thus, the following is a composition series of V : 0 = X0 ⊂ X1 ⊂ · · · ⊂ Xr = Y0 ⊂ Y1 ⊂ · · · ⊂ Ys = V

(1.27)

Conversely, assume that V has a composition series (1.25). Put Ui = f −1 (Vi ) and observe that Ui /Ui−1 ,→ Vi /Vi−1 via f ; so each factor Ui /Ui−1 is either 0 or irreducible (in fact, isomorphic to Vi /Vi−1 ). Therefore, deleting repetitions from the chain 0 = U0 ⊆ U1 ⊆ · · · ⊆ Ul = U if necessary, we obtain a composition series for U . Similarly, putting Wi = g(Vi ), each factor Wi /Wi−1 is a homomorphic image of Vi /Vi−1 , and so we may again conclude that Wi /Wi−1 is either 0 or irreducible. Thus, we obtain the desired composition series of W by deleting superfluous members from the chain 0 = W0 ⊆ W1 ⊆ · · · ⊆ Wl = W . This proves (a). In preparation for the proof of (b), let us also observe that if U 6= 0 or, equivalently, Ker g 6= 0, then some factor Wi /Wi−1 will definitely be 0 above. Indeed,

1.2 Representations

33

there is an i such that Ker g ⊆ Vi but Ker g * Vi−1 . By irreducibility of Vi /Vi−1 , it follows that Vi = Vi−1 + Ker g and so Wi = Wi−1 . Hence W has a composition series of shorter length than the given composition series of V . (b) We will argue by induction on `(V ), which we define to be the minimum length of any composition series of V . If `(V ) = 0, then V = 0 and the theorem is clear. From now on assume that V 6= 0. For each subrepresentation 0 6= U ⊆ V , the factor V /U also has a composition series by part (a) and the observation in the last paragraph of the proof of (a) tells us that `(V /U ) < `(V ). Thus, by induction, the theorem holds for all V /U with U 6= 0. Now consider two composition series as in the theorem. If V1 = V10 , then 0 = V1 /V1 ⊂ V2 /V1 ⊂ · · · ⊂ Vl /V1 = V /V1 and

(1.28)

0 = V10 /V1 ⊂ V20 /V1 ⊂ · · · ⊂ Vl00 /V1 = V /V1

are two composition series of V /V1 with factors isomorphic to Vi /Vi−1 (i = 0 (j = 2, . . . , l0 ), respectively. Thus the result follows from 2, . . . , l) and Vj0 /Vj−1 our inductive hypothesis on V /V1 in this case. So assume that V1 6= V10 and note that this implies V1 ∩ V10 = 0 by irreducibility of V1 and V10 . First, let us consider composition series for V /V1 ; one is already provided by (1.28). To build another, put U = V1 ⊕ V10 ⊆ V and fix a composition series for V /U , say 0 ⊂ U1 /U ⊂ · · · ⊂ Us /U = V /U . Then we obtain the following composition series for V /V1 : 0 ⊂ U/V1 ⊂ U1 /V1 ⊂ · · · ⊂ Us /V1 = V /V1

(1.29)

The first factor of this series is U/V1 ∼ and the remaining factors are isomorphic = to Ui /Ui−1 (i = 1, . . . , s), with U0 := U . By our inductive hypothesis on V /V1 , the collections of factors in (1.28) and (1.29), with multiplicities, are the same up to isomorphism. Adding V1 to both collections, we conclude that there is a bijective correspondence between the following two collections of irreducible representations, with corresponding representations being isomorphic: V10

Vi /Vi−1 (i = 1, . . . , l)

and

V1 , V10 , Ui /Ui−1 (i = 1, . . . , s)

Considering V /V10 in place of V /V1 , we similarly obtain a bijection between the 0 collection on the right and Vj0 /Vj−1 (j = 1, . . . , l0 ), which implies the theorem. t u The Jordan-Hölder Theorem allows us to define the length of any finite-length representation V ∈ Rep A by def

length V = the common length of all composition series of V If V has no composition series, then we put length V = ∞. Thus, length V = 0 means that V = 0 and length V = 1 says that V is irreducible. If V is a representation of a division algebra D, then length V = dimD V . In general, for any short

34

1 Representations of Algebras

exact sequence 0 → U → V → W → 0 in Rep A, we have the following generalization of a standard dimension formula for vector spaces (with the usual rules regarding ∞): length V = length U + length W (1.30) To see this, just recall that any two composition series of U and W can be spliced together to obtain a composition series for V as in (1.27). The Jordan-Hölder Theorem also tells us that, up to isomorphism, the collection of factors Vi /Vi−1 ∈ Irr A occuring in (1.25) is independent of the particular choice of composition series of V . These factors are called the composition factors of V . The number of occurrences, again up to isomorphism, of a given S ∈ Irr A as a composition factor in any composition series of V is also independent of the choice of series; it is called the multiplicity of S in V . We will write def

µ(S, V ) = multiplicity of S in V For any finite-length representation V , we evidently have X length V = µ(S, V ) S∈Irr A

and, by the same argument as above, (1.30) can be refined to the statement that multiplicities are additive in short exact sequences: for every S ∈ Irr A, µ(S, V ) = µ(S, U ) + µ(S, W )

(1.31)

1.2.5 Endomorphism Algebras and Schur’s Lemma The following general lemma describes the endomorphism algebras of irreducible representations. Although very easy, it will be of great importance in the following. Schur’s Lemma. For S ∈ Irr A, every nonzero morphism S → V in Rep A is injective and every nonzero morphism V → S is surjective. In particular, EndA (S) is a division k-algebra. If S is finite-dimensional, then EndA (S) is algebraic over k. Proof. If f : S → V is nonzero, then Ker f is a subrepresentation of S with Ker f 6= S. Since S is irreducible, it follows that Ker f = 0 and so f is injective. Similarly, for any 0 6= f ∈ Hom(V, S), we must have Im f = S, because Im f is a nonzero subrepresentation of S. It follows that any nonzero morphism between irreducible representations of A is injective as well as surjective, and hence it is an isomorphism. In particular, all nonzero elements of the algebra EndA (S) have an inverse, proving that EndA (S) is a division k-algebra. Finally, if S is finite-dimensional over k, then so is Endk (S). Hence, for each f ∈ Endk (V ), the powers f i (i ∈ Z+ ) are linearly dependent and so f satisfies a nonzero polynomial over k. Consequently, the division algebra EndA (S) is algebraic over k. t u

1.2 Representations

35

We will refer to EndA (S) as the Schur division algebra of the irreducible representation S and we will use the notation def

D(S) = EndA (S) We also define def

Irrfin A = {S ∈ Irr A | dimk S < ∞} Splitting fields and the weak Nullstellensatz Using the above notation, Schur’s Lemma asserts that D(S)/k is algebraic for all S ∈ Irrfin A. Algebras A such that D(S)/k is algebraic for all S ∈ Irr A are said to satisfy the weak Nullstellensatz. See the discussion in §5.6.1 and in Appendix C for the origin of this terminology. Thus, finite-dimensional algebras certainly satisfy the weak Nullstellensatz, because all their irreducible representations are finitedimensional (Lemma 1.14). The weak Nullstellensatz will later also be established, in a more laborious manner, for certain infinite-dimensional algebras (Section 5.6). Exercise 1.25 discusses a “quick and dirty” way to obtain the weak Nullstellensatz under the assumption that the cardinality of the base field k is larger than dimk A. In particular, if k is uncountable, then any affine k-algebra satisfies the weak Nullstellensatz. We will say that the base field k of a k-algebra A is a splitting field for A if D(S) = k for all S ∈ Irrfin A. By Schur’s Lemma, this certainly holds if k is algebraically closed, but sometimes much less is required. See Corollary 4.16 below for an important example. We will elaborate on the significance of the condition D(S) = k for arbitrary algebras in the next paragraph and again in Proposition 1.36 below. Centralizers and double centralizers We momentarily interrupt our focus on irreducible representations in order to introduce the double centralizer algebra of an arbitrary representation ρ : A → Endk (V ). Note that the endomorphism algebra EndA (V ) is the centralizer of AV = ρ(A) in Endk (V ): EndA (V ) = {f ∈ Endk (V ) | aV ◦ f = f ◦ aV for all a ∈ A} (1.20)

= CEndk (V ) (AV ) The centralizer of EndA (V ) in Endk (V ) is called the bi-commutant or double centralizer of the representation V : def

BiComA (V ) = CEndk (V ) (EndA (V )) = CEndk (V ) (CEndk (V ) (AV ))

36

1 Representations of Algebras

Viewing V as a representation of the algebra EndA (V ) via the inclusion EndA (V ) ,→ Endk (V ), we may also write BiComA (V ) = EndEndA (V ) (V )

(1.32)

Clearly, AV ⊆ BiComA (V ); so we may think of the given representation ρ as a map

∈

BiComA (V )

∈

ρ: A a

aV

Endk (V ) (1.33)

Evidently, BiComA (V ) = Endk (V ) if and only if EndA (V ) ⊆ Z (Endk (V )). Since Z (Endk (V )) = k, we obtain BiComA (V ) = Endk (V )

⇐⇒

EndA (V ) = k

(1.34)

Thus, k is a splitting field for A if and only if BiComA (S) = Endk (S) for all S ∈ Irrfin A. 1.2.6 Indecomposable Representations A nonzero V ∈ Rep A is said to be indecomposable if V cannot be written as a direct sum of nonzero subrepresentations. Irreducible representations are evidently indecomposable, but the converse is far from true. For example, Areg is indecomposable for any commutative domain A, because any two nonzero subrepresentations (ideals of A) have a nonzero intersection. If V is finite-dimensional, then V can be decomposed into a finite direct sum of indecomposable subrepresentations. Indeed, V = 0 is a direct sum with 0 indecomposable summands; and any 0 6= V ∈ Rep A is either already indecomposable or else V = V1 ⊕ V2 for nonzero subrepresentations Vi which both have a decomposition of the desired form by induction on the dimension. More interestingly, the decomposition of V thus obtained is essentially unique. This is the content of the following classical theorem, which is usually attributed to Krull and Schmidt. Various generalizations of the result also have the names of Remak and/or Azumaya attached, but we shall focus on the case of finite-dimensional representations here. Krull-Schmidt Theorem. Any finite-dimensional representation of an algebra can be decomposed into a finite direct sum of indecomposable subrepresentations and this decomposition is unique up to the order of the summands and up to isomorphism. Lr Ls More explicitly, the uniqueness statement asserts that if i=1 Vi ∼ = j=1 Wj for finite-dimensional indecomposable Vi , Wj ∈ Rep A, then r = s and there is a permutation s of the indices such that Vi ∼ = Ws(i) for all i . The proof will depend on the following lemma. Lemma 1.19. Let V ∈ Rep A be a finite-dimensional and indecomposable. Then each φ ∈ EndA (V ) is either an automorphism or nilpotent. Furthermore, the nilpotent endomorphisms form an ideal of EndA (V ).

1.2 Representations

37

Proof. Viewing V as a representation of the polynomial algebra k[t], with t acting as φ, we know from the Structure Theorem for Modules over PIDs (e.g., [55, Chapter 12]) that V is the direct sum of various primary components {v ∈ V | p(φ)r (v) = 0 for some r ∈ Z+ } for irreducible polynomials p(t) ∈ k[t]. Note that each primary component is an Asubrepresentation of V . Since V is assumed indecomposable, there can only be one nonzero component. Thus, p(φ)r = 0 for some monic irreducible p(t) ∈ k[t] and some r ∈ Z+ . If p(t) = t then φr = 0. If p(t) 6= t, then 1 = ta(t) + p(t)r b(t) for some a(t), b(t) ∈ k[t] and it follows that a(φ) = φ−1 . This proves the first assertion. For the second assertion, consider φ, ψ ∈ EndA (V ). If φ ◦ ψ is an automorphism of V , then so are both φ and ψ. Thus, we only need to show that if φ, ψ are nilpotent, then θ = φ + ψ is nilpotent as well. But otherwise θ is an automorphism and IdV −θ−1 ◦P φ = θ−1 ◦ ψ. The right hand side is nilpotent while the left hand side has inverse i≥0 (θ−1 ◦ φ)i , giving the desired contradiction. t u The ideal N = {φ ∈ EndA (V ) | φ is nilpotent} in Lemma 1.19 is clearly the unique largest left and right ideal of EndA (V ) and EndA (V )/N is a division algebra. Thus, the algebra EndA (V ) is local. With the lemma in hand, we can now give the Proof of the Krull-Schmidt Theorem. Only Lr Lsuniqueness still needs to be addressed. ∼ So let φ : V := V −→ W := i=1 i j=1 Wj be an isomorphism, with finitedimensional indecomposable representations Vi and Wj . Let µi : Vi ,→ V be the embedding into the ith component and πi : V  Vi the projection map; so πj ◦ P µi = δi,j IdVi and i πi ◦ µi = IdV . Similarly, we also have µ0j : Wj ,→ W and πj0 : W  Wj . Consider the maps αj := πj0 ◦ φ ◦ µ1 : V1 −→ Wj and βj := P π1 ◦ φ−1 ◦ µ0j : Wj −→ V1 . Since j βj ◦ αj = IdV1 , it follows from Lemma 1.19 that some βj ◦ αj must be an automorphism of V1 ; after renumbering if necessary, we may assume that j = 1. Since W1 is indecomposable, it further follows that α1 and β1 are isomorphisms (Exercise 1.2); so V1 ∼ = W1 . Finally, consider the map L L ∼ α>1 : V>1 := i>1 Vi V W 0 W>1 := j>1 Wj µ>1

φ

π>1

0 where µ>1 and π>1 are the standard embedding and projection maps of the direct sum. It suffices to show that this map is injective; for, then it must be an isomorphism for dimension reasons and we may apply induction to finish the proof. So let v ∈ Ker α>1 . Then φ ◦ µ>1 (v) = µ01 (w) for some w ∈ W1 . Since β1 is mono and β1 (w) = π1 ◦ φ−1 ◦ φ ◦ µ>1 (v) = 0, it follows that w = 0, and since φ ◦ µ>1 is mono as well, it further follows that v = 0 as desired. t u

Exercises Unless mentioned otherwise, A denotes an arbitrary k-algebra in these exercises.

38

1 Representations of Algebras

1.16 (Kernels). Given a k-algebra map φ : A → B, consider the restriction and B induction functors ResB A and IndA as defined in §1.2.2. Show: −1 (a) If V ∈ Rep B, then Ker(ResB (Ker V ). AV)=φ (b) Let W ∈ Rep A and assume that B is free as right A-module via φ. Then Ker(IndB A W ) = {b ∈ B | bB ⊆ Bφ(Ker W )}; this is the largest ideal of B that is contained in the left ideal Bφ(Ker W ). 1.17 (Faithful representations). Let V ∈ Repfin A. Show that V is faithful if and only if Areg is a subrepresentation of V ⊕n for some n ∈ N. 1.18 (Twisting representations). For V ∈ Rep A and α ∈ AutAlgk (A), recall the construction of the twisted representation α V from (1.23). Show: α◦β (a) α (β V ) ∼ V for all α, β ∈ AutAlgk (A). = (b) If α ∈ AutAlgk (A) is an inner automorphism, say α(a) = uau−1 for u ∈ A× , α then α V ∼ = V in Rep A via v ↔ u.v. α ∼ α (c) The map : V −→ V yields a bijection between the subrepresentations of V and α V . In particular, α V is irreducible, completely reducible, has finite length etc. if and only if this holds for V . α (d) α Areg ∼ = Areg via a ↔ α(a). 1.19 (Extension of scalars for homomorphisms). For given representations V, W ∈ Rep A and a given field extension K/k, show that the K-linear map (B.26) restricts to a K-linear map

∈

HomK⊗A (K ⊗ V, K ⊗ W )

∈

K ⊗ HomA (V, W ) λ⊗f

ρreg (λ) ⊗ f

Use the facts stated in §B.3.4 to show that this map is always injective and that it is bijective if V is finite-dimensional or the field extension K/k is finite. 1.20 (Noether-Deuring Theorem). Let V, W ∈ Repfin A and let K/k be a field extension. Then K ⊗ V ∼ = K ⊗ W in Rep (K ⊗ A) if and only if V ∼ = W in Rep A. To prove the nontrivial direction, assume that K ⊗ V ∼ = K ⊗ W in Rep (K ⊗ A) and complete the following steps. t (a) Fix a k-basis P{φi }1 of HomA (V, W ) and write each Φ : K ⊗ V → K ⊗ W uniquely as Φ = i λi ⊗ φi with λi ∈ K (Exercise 1.19). Show that the determinant det Φ = Λn Φ (n = dimk V = dimk W ) is a homogeneous polynomial f (λ1 , . . . , λt ) of degree n over k and f (λ1 , . . . , λt ) 6= 0 for some (λi ) ∈ K d . 0 0 0 d (b) If |k| > n,Pconclude that f (λP 1 , . . . , λt ) 6= 0 for some (λi ) ∈ k . Conclude 0 0 0 further that Φ = i λi ⊗ φi = 1 ⊗ i λi φi ∈ HomA (V, W ) is an isomorphism. (c) If |k| ≤ n, then choose some finite field extension F/k with |F | > n and elements µi ∈ F with f (µ1 , . . . , µt ) 6= 0 to obtain F ⊗ V ∼ = F ⊗ W . Conclude that ⊕d V ⊕d ∼ with d = [F : k] and invoke the Krull-Schmidt Theorem (§1.2.6) to =W further conclude that V ∼ = W in Rep A.

1.2 Representations

39

1.21 (Cofinite subalgebras). Let A be a k-algebra and let B be a subalgebra such that A is finitely generated as a left B-module, say A = Ba1 + · · · + Bam . (a) Show that, for any W
in Rep B, there is a k-linear embedding CoindA B W ,→ ⊕m W given by f 7→ f (ai ) . (b) Let 0 6= V ∈ Rep A be finitely generated. Use Exercise 1.3(a) to show that, for some W ∈ Irr B, there is a surjective homomorphism of representations, ResA B V  W. (c) Conclude from (a), (b) and Proposition 1.9 that, for every V ∈ Irr A, there exists some W ∈ Irr B such that V embeds into W ⊕m as a k-vector space. 1.22 (Representations of the Weyl algebra). Let A = A1 (k) denote the Weyl algebra; see (1.14). (a) Show that the standard representation of A (Example 1.13) is not equivalent to the regular representation Areg . Show also that the standard representation is faithful if char k = 0 (and recall from Example 1.13 that it is also irreducible in this case), but it is neither irreducible nor faithful if char k = p > 0; determine the kernel in this case. (b) Show that A has no nonzero finite-dimensional representations if char k = 0. (c) Assuming k to be algebraically closed with char k = p > 0, show that all irreducible representations of A have degree p. 1.23 (Finite length and chain conditions). Before tackling this exercise, it may be useful to review Exercise 1.4, which considers noetherian representations and the Ascending Chain Condition (ACC). A representation V ∈ Rep A is said to be artinian if V satisfies the Descending Chain Condition (DCC) on its subrepresentations: given any sequence U1 ⊇ U2 ⊇ U3 ⊇ . . . of subrepresentations of V , there exists some n such that Un = Un+1 = . . . . (a) Show that DCC is equivalent to the Minimum Condition on submodules: Every nonempty collection of submodules of V has at least one minimal member. (b) Given a short exact sequence 0 → U → V → W → 0 in Rep A, show that V is artinian if and only if both U and W are so. (c) Show that a representation has finite length if and only if it is artinian and noetherian. Give another proof of Theorem 1.18(a) using this fact in conjunction with (b) and Exercise 1.4(b). 1.24 (Finite length and filtrations). Let V ∈ Rep A. A filtration of length l of V , by definition, is any chain of subrepresentations F : 0 = V0 $ V1 $ · · · $ Vl = V . If all Vi also occur in another filtration of V , then the latter filtration is called a refinement of F ; the refinement is said to be proper if it has larger length than F . Thus, a composition series of V is the same as a filtration of finite length that admits no proper refinement. Prove: (a) If V has finite length, then any filtration F can be refined to a composition series of V . (b) V has finite length if and only if there is a bound on the lengths of all finitelength filtrations of V .

40

1 Representations of Algebras

1.25 (Weak Nullstellensatz for large base fields). Consider the Schur division algebra D(S) for S ∈ Irr A. (a) Show that dimk D(S) ≤ dimk S ≤ dimk A. (b) Show that, for any division k-algebra D and any d ∈ D that is not algebraic over k, the set {(d − λ)−1 | λ ∈ k} is linearly independent over k. (c) Conclude from (a) and (b) that if the cardinality |k| is strictly larger than dimk A, then D(S) is algebraic over k.

1.3 Primitive Ideals For many algebras A, the investigation of the set Irr A of irreducible representations benefits from an ideal theoretic perspective. The link between representations and ideals of A is provided by the notion of the kernel of a representation V ∈ Rep A, Ker V = {a ∈ A | a.v = 0 for all v ∈ V } Isomorphic representations evidently have the same kernel, but the converse is generally far from true. For example, the standard representation and the regular representation of the Weyl algebra are are not isomorphic (in any characteristic), even though they both have kernel (0) in characteristic 0 (Exercise 1.22). The kernels of irreducible representations of A are called the primitive ideals 2 of A. If S ∈ Irr A is written in the form S ∼ = A/L for some maximal left ideal L of A as in Lemma 1.14, then Ker S = {a ∈ A | aA ⊆ L} can also be described as the largest ideal of A that is contained in L. We shall denote the collection of all primitive ideals of A by Prim A Thus, we always have a surjective map

∈

Prim A

∈

Irr A S

Ker S

(1.35)

While this map is not bijective in general, the set Prim A does at least afford us a rough classification of the irreducible representations of A. 1.3.1 Degree-1 Representations Representations of degree 1 of any algebra A are clearly irreducible. They are given by homomorphisms φ ∈ HomAlgk (A, k), since Endk (V ) = k if dimk V = 1. For any such φ, we will use the notation 2

Strictly speaking, primitive ideals should be called left primitive, since irreducible representations are irreducible left modules. Right primitive ideals, defined as the annihilators of irreducible right modules, are not always primitive in the above sense [12].

1.3 Primitive Ideals

41

kφ to denote the field k with A-action a.λ = φ(a)λ for a ∈ A and λ ∈ k. The primitive ideal that is associated to the irreducible representation kφ is Ker kφ = Ker φ; this is an ideal of codimension 1 in A and all codimension-1 ideals have the form Ker φ with φ ∈ HomAlgk (A, k). Viewing k ⊆ A via the unit map, we have A = k ⊕ Ker φ and φ(a) is the projection of a ∈ A onto the first summand. Thus, we can recover φ from Ker φ. Consequently, restricting (1.35) to degree-1 representations, we obtain bijections of sets

⊆

Irr A

codimension-1 ideals of A

o

∼

n

equivalence classes of o degree-1 representations of A

∈

n

∈

∼

∈

HomAlgk (A, k) φ

Ker φ

kφ

(1.36)

1.3.2 Commutative Algebras If A is a commutative k-algebra, then maximal left ideals are the same as maximal ideals of A. Thus, denoting the collection of all maximal ideals of A by MaxSpec A ∼ A/P for some we know from Lemma 1.14 that each S ∈ Irr A has the form S = P ∈ MaxSpec A. Since Ker(A/I) = I holds for every ideal I of A, we obtain that P = Ker S and S ∼ = A/ Ker S. This shows that the primitive ideals of A are exactly the maximal ideals and that (1.35) is a bijection for commutative A: Prim A = MaxSpec A ∈

∼

∈

Irr A A/P

P

(1.37)

Thus, for commutative A, the problem of describing Irr A reduces to the description of MaxSpec A. Now assume that A is affine commutative and that the base field k is algebraically closed. Then all irreducible representations A/P are 1-dimensional by Hilbert’s Nullstellensatz (Appendix C). Hence, for any P ∈ Spec A, the following are equivalent: P is primitive ⇐⇒ P is maximal ⇐⇒ A/P = k (1.38) In view of (1.36) we obtain a bijection of sets ∼ Irr A ←→ HomAlgk (A, k)

(1.39)

42

1 Representations of Algebras

With this identification, Irr A can be thought of geometrically as the set of closed points of an affine algebraic variety over k. For example, (1.7) tells us that, for the polynomial algebra k[x1 , x2 , . . . , xn ], the variety in question is affine n-space kn : ∼ Irr k[x1 , x2 , . . . , xn ] ←→ kn

The pull-back of an irreducible representation of A along any k-algebra map φ : B → A, as in §1.2.2, is a degree-1 representation of B; so we obtain a map φ∗ = ResA B : Irr A → Irr B. If B is also affine commutative, then this is a morphism of affine algebraic varieties [81]. These remarks place the study of irreducible representations of affine commutative algebras over an algebraically closed base field within the realm of algebraic geometry which is outside the scope of this book. Nonetheless, the geometric context sketched above does provide the original background for some of the material on primitive ideals to be discussed later in this section. We end our excursion on commutative algebras with a simple example. Example 1.20. As was mentioned, the irreducible representations of the polynomial algebra A = k[x, y] over an algebraically closed field k correspond to the points of the affine plane k2 . Let us consider the subalgebra B = k[x2 , y 2 , xy] and let φ : B ,→ A denote the inclusion map. It is not hard to see that B ∼ = k[x1 , x2 , x3 ]/(x1 x2 − x23 ); so the irreducible representations of B correspond to the points of the cone x23 = x1 x2 in k3 . The following picture illustrates the restriction map φ∗ = ResA B : Irr A → Irr B; this map is easily seen to be surjective. k[x, y]

3

Out[7]=

(λ, µ)

∗

φ

φ

k[x2 , y 2 , xy]

2

2

3 (λ , µ , λµ)

1.3.3 Connections with Prime and Maximal Ideals For a general algebra A, primitive ideals are sandwiched between maximal and prime ideals of A: MaxSpec A ⊆ Prim A ⊆ Spec A

(1.40)

Here, MaxSpec A is the set of all maximal ideals of A as in §1.3.2 and Spec A denotes the set of all prime ideals of A. Recall that an ideal P of A is prime if P 6= A and IJ ⊆ P for ideals I, J of A implies that I ⊆ P or J ⊆ P . To see that primitive ideals are prime, assume that P = Ker S for S ∈ Irr A and let I, J be ideals of A such that I * P and J * P . Then I.S = S = J.S by irreducibility,

1.3 Primitive Ideals

43

and hence IJ.S = S. Therefore, IJ * P as desired. For the first inclusion in (1.40), let P ∈ MaxSpec A and let L be any maximal left ideal of A containing P . Then A/L is irreducible and Ker(A/L) = P . Thus, all maximal ideals of A are primitive, thereby establishing the inclusions in (1.40). As we shall see, these inclusions are in fact equalities if the algebra A is finite-dimensional (Theorem 1.34). However, in general, all inclusions in (1.40) are strict; see Example 1.22 below and many others later on in this book. We also remind the reader that an ideal I of A is called semiprime if, for any ideal J of A and any non-negative integer n, the inclusion J n ⊆ I implies that J ⊆ I. Prime ideals are clearly semiprime and intersections of semiprime ideals are evidently semiprime again. Thus, the intersection of any collection of primes is a semiprime ideal. In fact, by a standard ring theoretic result, all semiprime ideals arise in this manner: semiprime ideals are exactly the intersections of collections of primes (e.g., [107, 10.11]). 1.3.4 The Jacobson-Zariski Topology The set of Spec A of all prime ideals of an arbitrary algebra A carries a useful topology, the Jacobson-Zariski topology; it is also sometimes referred to under just one of these names or as Stone topology. This topology is defined by declaring the subsets of the form def V (I) = {P ∈ Spec A | P ⊇ I} to be closed; S here, I can T be any subset of A. Evidently, V (∅) = Spec A, V ({1}) = ∅ and V ( α Iα ) = α V (Iα ) for any collection {Iα } of subsets of A. Moreover, we may clearly replace a subset I ⊆ A by the ideal of A that is generated by I without changing V (I). Thus, the closed subsets of Spec A can also be described as the sets of the form V (I), where I is an ideal of A. The defining property of prime ideals implies that V (I) ∪ V (J) = V (IJ) for ideals I and J. Thus, finite unions of closed sets are again closed, thereby verifying the topology axioms. The JacobsonZariski topology on Spec A induces a topology on the subset Prim A, the closed subsets being those of the form V (I) ∩ Prim A, and similarly for MaxSpec A. The Jacobson-Zariski topology is related to the standard Zariski topology on a finite-dimensional k-vector space V ; see Appendix C. Indeed, let O(V ) = Sym V ∗ denote the algebra of polynomial functions on V (Section C.3). If k is algebraically closed, then the weak Nullstellensatz (Section C.1) yields a bijection MaxSpec O(V ) ∈

∼

∈

V v

mv := {f ∈ O(V ) | f (v) = 0}

Viewing this as an identification, the Zariski topology on V is readily seen to coincide with the Jacobson-Zariski topology on MaxSpec O(V ).

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1 Representations of Algebras

In comparison with the more familiar topological spaces from analysis, say, the topological space Spec A for an algebra A generally has rather bad separation properties. Indeed, a “point” P ∈ Spec A is closed exactly if the prime ideal P is in fact maximal. Exercise 1.27 explores the Jacobson-Zariski topology in some more detail. Here, we content ourselves by illustrating it with two examples: polynomial algebras and the so-called quantum plane (Exercise 1.15). Further examples will follow later. Example 1.21 (Polynomial algebras). Starting with k[x], we have  Spec k[x] = (0) t MaxSpec k[x]   = (0) t (f ) | f ∈ k[x] irreducible If k is algebraically closed, then MaxSpec k[x] is in bijection with k via (x−λ) ↔ λ. Therefore, one often visualizes Spec k[x] as a “line”, the points on the line corresponding to the maximal ideals and the line itself corresponding to the ideal (0). The latter ideal is a generic point for the topological space Spec A in the sense that the closure of (0) is all of Spec A. Figure 1.1 renders Spec k[x] in three ways, with red dots representing maximal ideals in each case. The solid gray lines in the top picture represent inclusions. The large black area in the other two pictures represents the generic point (0). The third picture also aims to convey the fact that (0) is the determined by the maximal ideals, being their intersection, and that the topological space Spec k[x] is quasi-compact (Exercise 1.27). ...

. . . (x − λ)

(0)

Fig. 1.1. Spec k[x]

The situation becomes somewhat more difficult to visualize for k[x, y]. As a set, we have   Spec k[x, y] = (0) t (f ) | f ∈ k[x, y] irreducible t MaxSpec k[x, y] Assuming k to be algebraically closed, maximal ideals of k[x, y] are in bijection with points of the plane k2 via (x − λ, y − µ) ↔ (λ, µ). Figure 1.2 depicts the topological space Spec k[x, y], the generic point (0) again being represented by a large black region. The two curves in the plane are representative for the infinitely many primes that are generated by irreducible polynomials f ∈ k[x, y]; and finally, we have sprinkled a few red points throughout the plane to represent MaxSpec k[x, y]. A point lies on a curve exactly if the corresponding maximal ideal contains the principal ideal (f ) giving the curve.

1.3 Primitive Ideals

45

Fig. 1.2. Spec k[x, y]

Example 1.22 (The quantum plane). Fix a scalar q ∈ k× that is not a root of unity. The quantum plane is the algebra
A = Oq (k2 ) = khx, yi/ xy − qyx Our goal is to describe Spec A, paying attention to which primes are primitive or maximal. First note that the zero ideal of A is certainly prime, because A is a domain by Exercise 1.15(a). It remains to describe the nonzero primes of A. We refer to Exercise 1.15(c) for the fact that every nonzero ideal of A contains some standard monomial xi y j . Observe that both x and y are normal elements of A in the sense that (x) = xA = Ax and similarly for y. Therefore, if xi y j ∈ P for some P ∈ Spec A, then xi y j A = (x)i (y)j ⊆ P , and hence x ∈ P or y ∈ P . In the former case, P/(x) is a prime ideal of A/(x) ∼ = k[y], and hence P/(x) is either the zero ideal or else P/(x) is generated by some
irreducible polynomial g(y). Thus, if x ∈ P either P = (x) or P = x, g(y) , which
is maximal. Similarly, if y ∈ P either P = (y) or P is the maximal ideal y, f (x) for some irreducible f (x) ∈ k[x]. Only (x, y) occurs in both collections of primes, corresponding to g(y) = y or f (x) = x. Therefore Spec A can be pictured as shown in Figure 1.3. Solid gray lines represent inclusions as in Figure 1.1, and primitive ideals are marked in red. The maximal ideals on top of the diagram in Figure 1.3 are all primitive by (1.40). On the other hand, neither (x) nor (y) are primitive by (1.37), because they correspond to nonmaximal ideals of commutative (in fact, polynomial) algebras. It is less clear, why the zero ideal should be primitive. The reader is asked to verify this in Exercise 1.30, but we will later see (Exercise 5.32) that primitivity of (0) also follows from the fact that the intersection of all nonzero primes is nonzero, which is clear from Figure 1.3: (x) ∩ (y) 6= (0). Note that, in this example, all inclusions in (1.40) are strict.

46

1 Representations of Algebras 

x, g(y)






y, f (x)

...


...

x, y

(x)




(y)

(0) 2

Fig. 1.3. Spec Oq (k ) (q not a root of unity)

We finish our discussion of the quantum plane by offering another visualization of Spec A, which emphasizes the fact that (0) is a generic point for the topological space Spec A — this point is represented by the large red area in the picture on the right. We also assume k to be algebraically
closed. The maximal ideals x, g(y) = (x, y − η) with η ∈ k are represented by points on the y-axis, the axis itself being the generic point (x), and similarly for the x-axis with generic point (y).

1.3.5 The Jacobson Radical The intersection of all primitive ideals of an arbitrary algebra A will play an important role in the following; it is called the Jacobson radical of A: def

rad A =

\

 P = a ∈ A | a.S = 0 for all S ∈ Irr A

P ∈Prim A

Being an intersection of primes, the Jacobson radical is a semiprime ideal of A. Algebras with vanishing Jacobson radical are called semiprimitive. Since (rad A).S = 0 holds for all S ∈ Irr A, inflation along the canonical map A  A/ rad A as in §1.2.2 yields a bijection ∼ Irr(A/ rad A) ←→ Irr A

(1.41)

Moreover, primitive ideals of A are in one-to-one correspondence with those of A/ rad A via P ↔ P/ rad A. Therefore,

1.3 Primitive Ideals

rad(A/ rad A) = 0

47

(1.42)

We finish this section by giving a purely ring theoretic description of the Jacobson radical in the case of a finite-dimensional algebra A which makes no mention of representations: rad A is the largest nilpotent ideal of A. Recall that an ideal I of an algebra A is called nilpotent if I n = 0 for some n; similarly for left or right ideals. Proposition 1.23. The Jacobson radical rad A of any algebra A contains all nilpotent left and right ideals of A. Moreover, for each finite-length V ∈ Rep A, (rad A)length V .V = 0 If A is finite-dimensional, then rad A is itself nilpotent. Proof. We have already pointed out earlier that rad A is a semiprime ideal of A. Now, any semiprime ideal I of A contains all left ideals L of A such that Ln ⊆ I. To see this, note that LA is an ideal of A that satisfies (LA)n = Ln A ⊆ I. By the defining property of semiprime ideals, it follows that LA ⊆ I and hence L ⊆ I. A similar argument applies to right ideals. In particular, every semiprime ideal contains all nilpotent left and right ideals of A. This proves the first statement. Now assume that 0 = V0 ⊂ V1 ⊂ · · · ⊂ Vl = V is a composition series of V . Since rad A annihilates all irreducible factors Vi /Vi−1 , it follows that (rad A).Vi ⊆ Vi−1 and so (rad A)l .V = 0. For finite-dimensional A, this applies to the regular representation V = Areg , giving (rad A)l .Areg = 0 and hence (rad A)l = 0. t u

Exercises In these exercises, A denotes an arbitrary k-algebra unless specified otherwise. 1.26 (Commutative cofinite subalgebras). Let A be an affine k-algebra having a commutative subalgebra B ⊆ A such that A is finitely generated as left B-module. Use the weak Nullstellensatz (Appendix C), the Artin-Tate Lemma (Exercise 1.7) and Exercise 1.21(c) to show that all irreducible representations of A are finitedimensional. 1.27 (Jacobson-Zariski topology). For any subset X ⊆ Spec A, put def

I (X) =

\

P

P ∈X

This is a semiprime ideal of A and all semiprime ideals of A are of this form. We also define the semiprime radical of any ideal I of A by √

def

I = I (V (I)) =

\ P ∈ Spec A P ⊇I

P

48

1 Representations of Algebras

This is clearly the smallest semiprime ideal of A containing I. (a) Show that the closure of any subset X ⊆ Spec A in the Jacobson-Zariski topology is given by X = V (I (X)). (b) Conclude that the following are inclusion reversing bijections that are inverse to each other: V (.) n o n o closed subsets of Spec A semiprime ideals of A I (.)

Thus, the Jacobson-Zariski topology on Spec A determines all semiprime ideals of A and their inclusion relations among each√ other. (c) Show that I (X) = I (X) and V ( I) = V (I) holds for any X ⊆ Spec A and any ideal I of A. (d) A topological space is said to be irreducible if it cannot be written as the union of two proper closed subsets. Show that, under the bijection in (b), the irreducible closed subsets of Spec A correspond to the prime ideals S of A. (e) Show that Spec A is quasi-compact: if Spec A = i∈I Ui for some collection S of open subsets of Ui ⊆ Spec A, then Spec A = i∈I 0 Ui with I 0 ⊆ I finite. 1.28 (Maximum condition on semiprime ideals). Assume that the algebra A satisfies the maximum condition on semiprime ideals: Every nonempty collection of semiprime ideals of A has at least one maximal member. Clearly, every right or left noetherian algebra satisfies this condition. Furthermore, affine PI-algebras are also known to satisfy MAXs.prime (e.g., Rowen [156, 6.3.36’]). (a) Show that every semiprime ideal of A is an intersection of finitely many primes of A. (b) Conclude that every closed subset of Spec A, for the Jacobson-Zariski topology, is a finite union of irreducible closed sets (Exercise 1.27). Moreover, the topology of Spec A is determined by the inclusion relations among the primes of A. MAXs.prime :

1.29 (Characterization of the Jacobson radical). Let A be an arbitrary k-algebra. Show that the following subsets of A are all equal to rad A: (i) the intersection of all maximal left ideals of A, (ii) the intersection of all maximal right ideals of A, (iii) the set {a ∈ A | 1 + xay ∈ A× for all x, y ∈ A}. 1.30 (Quantum plane). Let A = Oq (k2 ) be the quantum plane, with q ∈ k× not a root of unity. (a) Show that V = A/A(xy − 1) is a faithful irreducible representation of A. Thus, (0) is a primitive ideal of A. (b) Assuming k to be algebraically closed, show the following account for all closed subsets of Spec A: all finite subsets of MaxSpec A (including ∅), V (x) ∪ X for any finite subset X ⊂ {(x − ξ, y) | ξ ∈ k× }, V (y) ∪ Y for any finite subset Y ⊂ {(x, y − η) | η ∈ k× }, V (x) ∪ V (y), and Spec A. Here, we have written V (f ) = V ({f }) for f ∈ A.

1.4 Semisimplicity

49

1.31 (Centralizing homomorphisms). An algebra map φ : A → B is called centralizing if the algebra B is generated by φ(A) and the centralizer CB (φ(A)) = {b ∈ B | bφ(a) = φ(a)b ∀a ∈ A}. Surjective algebra maps are clearly centralizing, but there are many others, e.g., the embedding of A into the polynomial algebra A[x]. (a) Show that composites of centralizing homomorphisms are again centralizing. (b) Let φ : A → B be centralizing. Show that φ(Z A) ⊆ Z B. Furthermore, for every ideal I of A, show that Bφ(I) = φ(I)B. Deduce the existence of a map Spec B → Spec A, P 7→ φ−1 (P ). 3

1.4 Semisimplicity In some circumstances, a given representation of an algebra A can broken down into irreducible building blocks in a better way than choosing a composition series. Specifically, V ∈ Rep A is called completely reducible if V is a direct sum of irreducible subrepresentations. It turns out that completely reducible representations share some important features with vector spaces: the existence of complements for subrepresentations and a meaningful notion of “dimension”, here called length. In this section, we give several equivalent characterizations of complete reducibility (Theorem 1.26); we describe a useful decomposition of completely reducible representations, the decomposition into homogeneous components (§1.4.2); and we determine the structure of the algebras A having the property that all V ∈ Rep A are completely reducible (Wedderburn’s Structure Theorem). Algebras with this property are called semisimple. Unless explicitly mentioned otherwise, A will continue to denote an arbitrary k-algebra in this section. 1.4.1 Completely Reducible Representations Recall that V ∈ Rep A is said to be completely reducible (sometimes also semisimple) if M V = Si i∈I

with irreducible subrepresentations Si ⊆ V . Thus, each v ∈ V can be uniquely P written as a sum v = i∈I vi with vi ∈ Si and vi = 0 for all but finitely many i ∈ I. The case V = 0 is included here, corresponding to the empty sum. Example 1.24 (Division algebras). Every representation V of a division algebra is completely reducible. Indeed, any choice of basis for V yields a decomposition of V as a direct sum of irreducible subrepresentations. 3

This fails for the embedding of A into the power series algebra AJxK: there are examples, due to George Bergman, of primes P ∈ Spec AJxK such that P ∩ A is not even semiprime [143, Example 4.2].

50

1 Representations of Algebras

Example 1.25 (Polynomial algebras). By (1.6) representations of the polynomial algebra A = k[x1 , x2 , . . . , xn ] are given by a k-vector space V and a collection of n pairwise commuting operators (xi )V ∈ Endk (V ). Assuming k to be algebraically closed, V is irreducible if and only if dimk V = 1 by (1.38). A completely reducible representation V of A is thus given by n simultaneously diagonalizable operators (xi )V ∈ Endk (V ), that is, V has a k-basis consisting of eigenvectors for all (xi )V . If V is finite-dimensional, then we know from linear algebra that such a basis exists if and only if the operators (xi )V commute pairwise and the minimal polynomial of each (xi )V is separable, that is, it has no multiple roots. Characterizations of complete reducibility Recall the following familiar facts from linear algebra: all bases of a vector space have the same cardinality; every generating set of a vector space contains a basis; and every subspace of a vector space has a vector space complement. The theorem below extends these facts to completely reducible representations of arbitrary algebras. Given a representation V and a subrepresentation U ⊆ V , a complement for U in V is a subrepresentation C ⊆ V such that V = U ⊕ C. L Theorem 1.26. (a) Let V ∈ Rep A be completely reducible, say V = i∈I Si for irreducible subrepresentations Si . Then the following are equivalent: (i) I is finite; (ii) V has finite length; (iii) V is finitely generated. L ∼ L In this case, |I| = length V . In general, if i∈I Si = j∈J Tj with irreducible Si , Tj ∈ Rep A, then |I| = |J|. (b) The following are equivalent for any V ∈ Rep A : (i) V is completely reducible; (ii) V is a sum (not necessarily direct) of irreducible subrepresentations; (iii) Every subrepresentation U ⊆ V has a complement. L Proof. (a) First assume that I is finite, say I = {1, 2, . . . , l}, and put Vi = j≤i Sj . Then 0 = V0 ⊂ V1 ⊂ · · · ⊂ Vl = V is a composition series of V , with factors Vi /Vi−1 ∼ = Si . Thus length V = l, proving the implication (i) ⇒ (ii). Furthermore, since irreducible modules are finitely generated (in fact, cyclic), (ii) always implies (iii), even when V is not necessarily completely reducible (Exercise 1.3). Now assume that VLis finitely generated, say V = Av1 + Av2 + · · · + Avt . For S each j, we t have vj ∈ S for some finite subset I ⊆ I. It follows that I = j i∈Ij i j=1 Ij is finite. This proves the equivalence of (i) – (iii) as well as the equality |I| = length V for finite I. Since property (ii) and the value of length V only depend on the isomorphism type of V andLare defined independently of I, we also obtain that |I| = |J| if I is finite and V ∼ = j∈J Tj with irreducible Tj ∈ Rep A. It remains to show that |I| = |J| also holds if I and J are infinite. Replacing each Tj byL its image in V under the given isomorphism, we may assume that Tj ⊆ V and V = j∈J Tj . Select elements 0 6= vj ∈ Tj . Then Tj =SAvj and so {vj }j∈J is a generating set for V . Exactly as above, we obtain that I = j∈J Ij for suitable finite

1.4 Semisimplicity

51

S subsets Ij ⊆ I . Since J is infinite, the union j∈J Ij has cardinality at most |J|; see [21, Cor. 3 on p. E III.49]. Therefore, |I| ≤ |J|. By symmetry, equality must hold. P (b) The implication (i) ⇒ (ii) is trivial. Now assume (ii), say V = i∈I Si with irreducible subrepresentations Si ⊆ V . Claim. Given L any subrepresentation U ⊆ V , there exists a subset J ⊆ I such that V = U ⊕ i∈J Si . L This will prove (iii), with C = i∈J Si . Taking U = 0 in the Claim, we also obtain (i). To prove the Claim, choose a subset J ⊆ I that is maximal with respect P to the property that the sum U + i∈J Si is direct. The existence of J is clear if I is finite; in general, existence follows by a straightforward LZorn’s Lemma argument. It suffices to show that the subrepresentation V 0 := U ⊕ i∈J Si is equal to V . If not, then Sk * V 0 for some k ∈ I. Since Sk is irreducible, this forces Sk ∩V 0 = 0, which P 0 in turn implies that the sum V + Sk = U + i∈J∪{k} Sj is direct, contradicting the maximality property of J. Therefore, V 0 = V , proving the Claim. Finally, let us derive (ii) from (iii). To this end, let S denote the sum of all irreducible subrepresentations of V . Our goal is to show that S = V . If not, then V = S ⊕ C for some nonzero subrepresentation C ⊆ V by virtue of (iii). To reach a contradiction, it suffices to show that every nonzero subrepresentation C ⊆ V contains an irreducible subrepresentation. For this, we may clearly replace C by Av for some 0 6= v ∈ C, and hence we may assume that C is cyclic. Another application of Zorn’s Lemma shows that there is a subrepresentation D ⊆ C such that C/D is irreducible (Exercise 1.3). Using (iii) to write V = D ⊕ E, we obtain that C = D ⊕ (E ∩ C). Hence, E ∩ C ∼ = C/D is an irreducible subrepresentation of C as desired. This proves (ii), thereby finishing the proof of the theorem. t u Corollary 1.27. Subrepresentations and homomorphic images of completely P reducible representations are completely reducible. More precisely, if V = i∈I Si for irreducible subrepresentations Si ⊆ V , then all subrepresentations and all hoL momorphic images of V are equivalent to direct sums i∈J Si for suitable J ⊆ I. P Proof. First consider an epimorphism f : V =  W in Rep A. By the i∈I Si L Claim in the proof of Theorem 1.26, we have V = Ker f ⊕ i∈J Si for some J ⊆ I. ∼L ∼ V / Ker f = S , which proves the statement about homomorphic Hence, W = i∈J i images. Finally, every subrepresentation U ⊆ V is in fact also a homomorphic image of V . Indeed, choosing a complement C for U in V , we obtain a projection map V = U ⊕ C  U. t u Theorem 1.26(a) L allows us to define the length of any completely reducible representation V = i∈I Si by def

length V = |I| This agrees with our more general definition of length in §1.2.4 if length V < ∞, and it refines the earlier definition for completely reducible representations V of

52

1 Representations of Algebras

infinite length. We shall however mostly be interested in completely reducible representations V having finite length. In this case most of the set theoretic calisthenics in the proof of Theorem 1.26 are unnecessary. 1.4.2 Socle and Homogeneous Components The sum of all irreducible subrepresentations of an arbitrary V ∈ Rep A, which already featured in the proof of Theorem 1.26(b), is called the socle of V . For a fixed S ∈ Irr A, we will also consider the sum of all subrepresentations of V that are equivalent to S; this is called the S-homogeneous component of V : def

⊆

soc V = the sum of all irreducible subrepresentations of V def V (S) = the sum of all subrepresentations U ⊆ V such that U ∼ =S

Thus, a representation V is completely reducible if and only if V = soc V . In general, soc V is the unique largest completely reducible subrepresentation of V , and it is the sum of the various homogeneous components V (S). We will see below that this sum is in fact direct. Of course, it may happen that soc V = 0. Indeed, Example 1.17 shows that this is the case for the regular representation of any domain that is not a division algebra. Example 1.28 (Weight spaces and eigenspaces). If S = kφ is a degree-1 representation of an algebra A, with φ ∈ HomAlgk (A, k) as in §1.3.1, then the Shomogeneous component V (kφ ) will be written as Vφ ; so def

Vφ = {v ∈ V | a.v = φ(a)v for all a ∈ A} If Vφ 6= 0, then φ is called a weight of the representation V and Vφ is called the corresponding weight space. In the special case where A = k[t] is the polynomial algebra, the map φ is determined by the scalar λ = φ(t) ∈ k and Vφ is the usual eigenspace for the eigenvalue λ of the endomorphism tV ∈ Endk (V ). The following proposition generalizes some standard facts about eigenspaces. Proposition 1.29. Let V ∈ Rep A. Then: L (a) soc V = S∈Irr A V (S) (b) If f : V → W is a morphism in Rep A, then f (V (S)) ⊆ W (S) for all S ∈ Irr A . (c) For any subrepresentation U ⊆ V , we have soc U = U ∩ soc V and U (S) = U ∩ V (S) for all S ∈ Irr A .

1.4 Semisimplicity

53

Proof. (a) We only need to show that the sum of all homogeneous components is direct, that is, X V (T ) = 0 V (S) ∩ T ∈Irr A T 6=S

for all S ∈ Irr A. Denoting the above intersection by X, we know by Corollary 1.27 that X is completely reducible. Moreover, each irreducible subrepresentation of X is equivalent to S and also to one of the representations T ∈ Irr A with T 6= S. Since there are no such irreducible representations, we must have X = 0. This proves (a). For (b) and (c), note that Corollary 1.27 also tells us that f (V (S)) and U ∩ V (S) are both equivalent to direct sums of copies of the representation S, which clearly implies the inclusions f (V (S)) ⊆ W (S) and U ∩V (S) ⊆ U (S). Since the inclusion U (S) ⊆ U ∩ V (S) is obvious, the proposition is proved. t u Multiplicities For any V ∈ Rep A and any S ∈ Irr A, we put def

m(S, V ) = length V (S) Thus,

⊕m(S,V ) V (S) ∼ =S

(1.43)

where the right hand side denotes the direct sum of m(S, V ) many copies of S, and Proposition 1.29(a) implies that M ⊕m(S,V ) M soc V = V (S) ∼ S (1.44) = S∈Irr A

S∈Irr A

The foregoing will be most important in the case of a completely reducible representation V . In this case, (1.44) shows that V is determined, up to equivalence, by the cardinalities m(S, V ). Any S ∈ Irr A such that m(S, V ) 6= 0 is called an irreducible constituent of V . If V is completely reducible of finite length, then each m(S, V ) is identical to the multiplicity µ(S, V ) of S in V as defined in §1.2.4. Therefore, m(S, V ) is also referred to as the multiplicity of S in V , even when V is completely reducible of infinite length. If V is a finite-length representation that is not necessarily completely irreducible, then m(S, V ) ≤ µ(S, V ) for all S ∈ Irr A. The following proposition expresses multiplicities as dimensions. Recall from Example 1.3 or §B.2.1 that, for any V, W ∈ Rep A, the vector space HomA (V, W ) is a (EndA (W ), EndA (V ))-bimodule via composition. In particular, for any S ∈ Irr A, we may regard HomA (V, S) as a left vector space over the Schur division algebra D(S) in this way, and HomA (S, V ) as a right vector space over D(S). Proposition 1.30. Let V ∈ Rep A be completely reducible of finite length. Then, for any S ∈ Irr A, m(S, V ) = dimD(S) HomA (V, S) = dimD(S) HomA (S, V )

54

1 Representations of Algebras

Proof. The functors HomA ( · , S) and HomA (S, · ) commute with finite direct sums; see (B.13). Therefore, ⊕m(S,V ) ⊕m(S,V ) HomA (V, S) ∼ , S) ∼ = HomA (V (S), S) ∼ = HomA (S = D(S) (1.43)

where the first isomorphism follows from Schur’s Lemma. Consequently, m(S, V ) = dimD(S) HomA (V, S). The verification of the second equality is analogous; an explicit isomorphism is given by V (S) ∈

∼

∈

HomA (S, V ) ⊗D(S) S f ⊗s

f (s) t u

1.4.3 Semisimple Algebras The algebra A is called semisimple if the following equivalent conditions are satisfied: (i) the regular representation Areg is completely reducible; (ii) all representations of A are completely reducible. Condition (ii) certainly implies (i). For the converse, note that every V ∈ Rep A is a homomorphic image of a suitable direct sum of copies of the regular representation Areg : any generating set {vi | i ∈ I} of V gives rise to an epimorphism A⊕I reg  V , P ⊕I (ai ) 7→ i ai vi . Now Areg is completely reducible by (i), being a direct sum of completely reducible representations, and Corollary 1.27 then further implies that V is completely reducible as well; in fact, V is isomorphic to a direct sum of certain irreducible constituents of Areg , possibly with multiplicities > 1. Thus, (i) and (ii) are indeed equivalent. Furthermore, property (ii) evidently passes to homomorphic images of A, and hence all homomorphic images of semisimple algebras are again semisimple As we have seen in Example 1.24, division algebras are semisimple. The main result of this section, Wedderburn’s Structure Theorem, gives a complete description of all semisimple algebras: they are exactly the finite direct products of matrix algebras over various division algebras. Here, the direct product t Y

Ai = A1 × · · · × At

i=1

of algebras Ai is understood to have component-wise scalar multiplication, addition and multiplication; e.g., (x1 , x2 , . . . , xt )(y1 , y2 , . . . , yt ) = (x1 y1 , x2 y2 , . . . , xt yt ) for all xi , yi ∈ Ai . The characterization in terms of matrix algebras over division

1.4 Semisimplicity

55

algebras shows in particular that the notion of “semisimplicity” is right-left symmetric; this is not immediately evident from the above definition, since representations are left modules. Before stating Wedderburn’s Structure Theorem in full, we give a description of the endomorphism algebra of any finitely generated completely reducible representation V . Recall from (1.44) that V can be uniquely written as ⊕m ⊕m ⊕m V ∼ = S1 1 ⊕ S2 2 ⊕ · · · ⊕ St t

(1.45)

for pairwise distinct Si ∈ Irr A and positive integers mi . Proposition 1.31. Let V be as in (1.45) and let D(Si ) = EndA (Si ) denote the Schur division algebras. Then EndA (V ) ∼ =

t Y

Matmi (D(Si ))

i=1

Proof. By Schur’s Lemma, HomA (Si , Sj ) = 0 for i 6= j. Therefore, putting Di = D(Si ), Lemma 1.4(a) gives an algebra isomorphism  

EndA (V )

∼ =

  Matm1 (D1 )             

0 Matm (D2 ) 2

..

.

0

Matm (Dt )

              

t

which is exactly what the proposition asserts.

t u

Wedderburn’s Structure Theorem. A k-algebra A is semisimple if and only if A is isomorphic to a finite direct product of matrix algebras over division k-algebras: A ∼ =

t Y

Matmi (Di )

i=1

In this case, the data on the right are determined by A as follows: • • • •

t = # Irr A , say Irr A = {S1 , S2 , . . . , St }; op Di ∼ = D(Si ) ; mi = m(Si , Areg ) = dimD(Si ) Si ; Matmi (Di ) ∼ = BiComA (Si ).

56

1 Representations of Algebras

Proof. First assume that A is semisimple. Since the regular representation Areg is generated by the identity element of A, it follows from (1.45) that ⊕m ⊕m ⊕m Areg ∼ = S1 1 ⊕ S2 2 ⊕ · · · ⊕ St t

with pairwise distinct Si ∈ Irr A and positive integers mi . We obtain algebra isomorphisms ∼ A EndA (Areg )op = Lemma 1.5(b)

t Y

∼ =

Proposition 1.31

∼ =

Lemma 1.5(a)

i=1 t Y

Matmi (D(Si ))op Matmi (Di )

i=1

with Di = D(Si )op . Here, we have tacitly used the obvious facts that (Aop )op ∼ =A and that · op commutes with direct products. Since opposite algebras of division algebras are clearly division algebras as well, we have shown that any semisimple algebra is isomorphic to a finite direct product of matrix algebras over division algebras. Conversely, assume that A∼ =

t Y

Ai

with Ai = Matmi (Di )

i=1

for division k-algebras Di and positive integers mi . The direct product structure Qt A∼ = i=1 Ai implies that Areg

∼ =

t M

A

ResAi (Ai )reg

i=1 A

where ResAi (Ai )reg is the inflation of the regular representation of Ai along the standard projection A  Ai . Similarly, inflation injects each Irr Ai into Irr A and yields a bijection t G Irr Ai Irr A ∼ = i=1

Finally, viewing any V ∈ Rep Ai as a representation of A by inflation, the subalgebras AV and (Ai )V of Endk (V ) coincide, and hence also EndA (V ) = EndAi (V ) and BiComA (V ) = BiComAi (V ). In light of these remarks, we may assume that A = Matm (D) for some division k-algebra D and we must show: A is semisimple; # Irr A = 1, say Irr A = {S}; op m = m(S, Areg ) = dimD(S) S; D ∼ = D(S) ; and A ∼ = BiComA (S). Let Lj ⊆ A denote the collection of all matrices such that nonzero matrix entries L can only occur in the j th column. Then each Lj is a left ideal of A and Areg = j Lj . Moreover, ⊕m ⊕m Lj ∼ = S := Dreg as left module over A, with A acting on Dreg by matrix multi⊕m plication as in Lemma 1.4(b). Therefore, Areg ∼ = S . Moreover, since the regular

1.4 Semisimplicity

57

representation of D is irreducible, it is easy to see that S ∈ Irr A (Exercise 1.33). This shows that Areg is completely reducible, with m = m(S, Areg ), and so A is semisimple. Since every representation of a semisimple algebra is isomorphic to a direct sum of irreducible constituents of the regular representation, as we have already remarked at the beginning of this subsection, we also obtain that Irr A = {S}. As for the Schur division algebra of S, we have ⊕m D(S) = EndA (Dreg )

∼ = Lemma 1.4(b)

EndD (Dreg )

∼ =

Dop

Lemma 1.5(b)

op ⊕m as a Consequently, D ∼ = D(S) and dimD(S) S is equal to the dimension of D right vector space over D, which is m . Finally, op ⊕m BiComA (S) = EndD(S) (S) ∼ = EndDop ((D )reg ) (1.32)

∼ =

Lemma 1.5

Matm (D) = A

This finishes the proof of Wedderburn’s Structure Theorem.

t u

1.4.4 Some Consequences of Wedderburn’s Structure Theorem First, for future reference, let us restate the isomorphism in Wedderburn’s Structure Theorem. Corollary 1.32. If A is semisimple, then there is an isomorphism of algebras ∼

Y

BiComA (S)

S∈Irr A

a

∈

∈

A

aS




Split semisimple algebras Recall from (1.34) that k is a splitting field for A if and only if BiComA (S) = Endk (S) for all S ∈ Irr A. In this case, the isomorphism in Corollary 1.32 takes the form Y Y A∼ Endk (S) ∼ Matdimk S (k) (1.46) = = S∈Irr A

S∈Irr A

and the algebra A is called split semisimple. Our next corollary records some important numerology that results from the isomorphism (1.46). For any algebra A, the k-subspace of A that is generated by the Lie commutators [a, b] = ab − ba with a, b ∈ A will be denoted by [A, A]. Corollary 1.33. Let A be a split semisimple k-algebra. Then: (a) # Irr A = P dimk A/[A, A] (b) dimk A = S∈Irr A (dimk S)2 (c) m(S, Areg ) = dimk S for all S ∈ Irr A.

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1 Representations of Algebras

Proof. Under the isomorphism (1.46), the subspace [A, A] ⊆ A corresponds to Q S∈Irr A [MatdS (k), MatdS (k)], where we have put dS = dimk S. Each of the subspaces [MatdS (k), MatdS (k)] is a proper subspace of MatdS (k), being contained in the kernel of the trace MatdS (k)  k. On the other hand, using the elementary matrices ei,j having a 1 in position (i, j) and 0s elsewhere, we can form the commutators [ei,i , ei,j ] = ei,j (i 6= j) and [ei,i+1 , ei+1,i ] = ei,i − ei+1,i+1 , which span a subspace of codimension 1 in MatdS (k). Thus, each [MatdS (k), MatdS (k)] has codimension 1, and hence dimk A/[A, A] is equal to the number of matrix components in (1.46), which in turn equals # Irr A by Wedderburn’s Structure Theorem. This proves (a). Part (b) is clear from (1.46). Finally, (c) follows from (1.46) and the statement about multiplicities in Wedderburn’s Structure Theorem. t u Primitive central idempotents For a semisimple algebra A,Qwe let e(S) ∈ A denote the element corresponding to (0, . . . , 0, IdS , 0, . . . , 0) ∈ S∈Irr A BiComA (S) under the isomorphism of Corollary 1.32. Thus, e(S)S 0 = δS,S 0 IdS (1.47) for S, S 0 ∈ Irr A. All e(S) belong to the center Z A and they satisfy X e(S)e(S 0 ) = δS,S 0 e(S) and e(S) = 1

(1.48)

S∈Irr A

The elements e(S) are called the primitive central idempotents of A. For any V ∈ Rep A, it follows from (1.47) that the operator e(S)V is the identity on the S-homogeneous component V (S) and it annihilates all other homogeneous compoL nents of V . Thus, the idempotent e(S)V is the projection of V = S 0 ∈Irr A V (S 0 ) onto V (S): M proj. (1.49) e(S)V : V = V (S 0 ) V (S) 0

S ∈Irr A

1.4.5 Finite-Dimensional Irreducible Representations In this subsection, we record some applications of the foregoing to representations of algebras A that are not a priori assumed to be semisimple. Our focus will now turn to finite-dimensional representations. For any k-algebra A, the full subcategory of Rep A whose objects are the finite-dimensional representations will be denoted by Repfin A Recall from (1.33) that the image AV = ρ(A) of every representation ρ : A → Endk (V ) is contained in the double centralizer BiComA (V ) ⊆ Endk (V ).

1.4 Semisimplicity

59

Burnside’s Theorem. Let A be an arbitrary k-algebra and let V ∈ Repfin A. Then V is irreducible if and only if EndA (V ) is a division algebra and AV = BiComA (V ). In this case, AV is isomorphic to a matrix algebra over the division algebra D(V )op . Proof. First, assume that the given representation V is irreducible. Then we know by Schur’s Lemma that D(V ) = EndA (V ) is a division algebra. In order to show that AV = BiComA (V ), we may replace A by A = A/ Ker V , because AV = AV and BiComA (V ) = BiComA (V ). It suffices to show that A is semisimple; for, then Corollary 1.32 will tell us that AV = BiComA (V ). Fix a k-basis {vi }n1 of V . Then {a ∈ A | a.vi = 0 for all i} = KerA (V ) = 0, and hence we have an embedding

∈

V ⊕n

∈

Areg a

a.vi




Since V ⊕n is completely reducible, Corollary 1.27 implies that Areg is completely reducible as well, proving that A is semisimple as desired. Conversely, assume that D = EndA (V ) is a division algebra and that AV = BiComA (V ). Recall from (1.32) that BiComA (V ) = EndD (V ). Thus, AV = EndD (V ) and it follows from Example 1.12 that V is an irreducible representation of AV . Hence V ∈ Irr A, proving the first assertion of Burnside’s Theorem. Finally, if V is irreducible, then V is a finite-dimensional left vector space over the Schur division algebra D(V ) = EndA (V ). Now Lemma 1.5 implies that BiComA (V ) = EndD(V ) (V ) is a matrix algebra over D(V )op , which completes the proof. t u For any algebra A, we put def

Speccofin A = {P ∈ Spec A | dimk A/P < ∞} and similarly for Primcofin A and MaxSpeccofin A. The next theorem shows that all three sets coincide and that they are in bijection with Irrfin A. Of course, if A is finitedimensional, then Irrfin A = Irr A, Speccofin A = Spec A etc. We will call ideals of finite codimension simply cofinite. Theorem 1.34. Let A be an arbitrary k-algebra. Then all cofinite prime ideals of A are maximal; so MaxSpeccofin A = Primcofin A = Speccofin A Moreover, there is a bijection

∈

∈

Irrfin A ∼ Speccofin A S

Ker S

60

1 Representations of Algebras

Proof. In view of the general inclusions MaxSpec A ⊆ Prim A ⊆ Spec A in (1.40), the claimed equality of sets MaxSpeccofin A = Primcofin A = Speccofin A will follow if we can show that any P ∈ Speccofin A is in fact maximal. For this, after replacing A by A/P , we may assume that A is a finite-dimensional prime algebra and we must show that A is simple. Choose a minimal nonzero left ideal L ⊆ A. Then L is a finite-dimensional irreducible representation of A. Furthermore, since A is prime and (Ker L)(LA) = 0, we must have Ker L = 0. Therefore, Burnside’s Theorem implies that A ∼ = AL is isomorphic to a matrix algebra over some division algebra, and hence A is simple (Exercise 1.13). For the asserted bijection with Irrfin A, note that an irreducible representation S is finite-dimensional if and only if Ker S has finite codimension in A. Therefore, the surjection Irr A  Prim A, S 7→ Ker S, in (1.35) restricts to a surjection Irrfin A  Speccofin A. In order to show that this map is also injective, let S, S 0 ∈ Irrfin A be such that Ker S = Ker S 0 . Then AS ∼ = AS 0 , and this algebra is isomorphic to a matrix algebra over some division algebra by Burnside’s Theorem. Since such algebras have only one irreducible representation up to equivalence by Wedderburn’s 0 Structure Theorem, we must have S ∼ t u =S. 1.4.6 Semisimplicity and the Jacobson radical The following theorem gives an ideal theoretic characterization of semisimplicity for finite-dimensional algebras. Theorem 1.35. The following are equivalent for a finite-dimensional algebra A: (i) A is semisimple; (ii) rad A = 0; (iii) A has no nonzero nilpotent right or left ideals. ⊕m ⊕m ⊕m Proof. If A is semisimple, then Areg ∼ = S1 1 ⊕S2 2 ⊕· · ·⊕St t with Si ∈ Irr A. Since (rad A).Si = 0 for all i, it follows that (rad A).Areg = 0 and so rad A = 0. Thus (i) implies (ii). In view of Proposition 1.23, (ii) and (iii)Tare equivalent. It remains to show that (ii) implies (i). So assume that rad TrA = S∈Irr A Ker S = 0. Since A is finite-dimensional, some finite intersection i=1 Ker Si must be 0. Since the various Ker Si are distinct maximal ideals of A by Theorem 1.34, the Chinese Remainder TheoremQ(e.g., [20, Proposition 9 on p. A I.104]) yields an isomorphism r ∼ of algebras A −→ i=1 A/ Ker Si . Burnside’s Theorem further tells us that each A/ Ker Si is a matrix algebra over a division algebra. Semisimplicity of A now follows from Wedderburn’s Structure Theorem. This proves the theorem. t u

For a finite-dimensional algebra A, we will write def

As.s = A/ rad A and refer to As.s as the semisimplification of A. Since rad As.s = 0 by (1.42), the algebra As.s is indeed semisimple by Theorem 1.35.

1.4 Semisimplicity

61

1.4.7 Absolutely Irreducible Representations Recall that the base field k of a k-algebra A, not necessarily semisimple or finitedimensional, is said to be a splitting field for A if D(S) = k for all S ∈ Irrfin A. We now discuss the relevance of this condition in connection with extensions of the base field (§1.2.2). Specifically, the representation V is called absolutely irreducible if K ⊗ V is an irreducible representation of K ⊗ A for every field extension K/k. Note that irreducibility of K ⊗ V for even one given field extension K/k certainly forces V to be irreducible, because any subrepresentation 0 $ U $ V would give rise to a subrepresentation 0 $ K ⊗ U $ K ⊗ V . Proposition 1.36. Let A be an arbitrary k-algebra and let S ∈ Irrfin A. Then S is absolutely irreducible if and only D(S) = k. Proof. First assume that D(S) = k. Then AS = Endk (S) by Burnside’s Theorem (§1.4.5). If K/k is any field extension, then the canonical map K ⊗ Endk (S) → EndK (K ⊗ S) is surjective (in fact, an isomorphism). Hence, K ⊗ ρ maps K ⊗ A onto EndK (K ⊗ S) and so K ⊗ S is irreducible by Example 1.12. Conversely, if S is absolutely irreducible and k is an algebraic closure of k, then k ⊗ S is a finite-dimensional irreducible representation of k ⊗ A. Hence Schur’s Lemma implies that D(k ⊗ S) = k. Since D(k ⊗ S) ∼ = k ⊗ D(S) (Exercise 1.19), we conclude that D(S) = k. t u Recall that, for any S ∈ Irr A and V ∈ Rep A, the multiplicity m(S, V ) is defined as the length of the homogeneous component V (S). Corollary 1.37 (Frobenius Reciprocity). Let φ : A → B be a homomorphism of semisimple k-algebras and let S ∈ Irrfin A and T ∈ Irrfin B be absolutely irreducible. Then: B m(S, ResB A T ) = m(T, IndA S) Proof. Put V = IndB A S. Then V is completely reducible of finite length, and so Proposition 1.30 gives m(T, V ) = dimk HomB (V, T ), because D(T ) = k. Similarly, putting W = ResB A T , we obtain m(S, W ) = dimk HomA (S, W ). Finally, HomB (V, T ) ∼ t u = HomA (S, W ) as k-vector spaces by Proposition 1.9.

Exercises In all these exercises, A denotes a k-algebra. 1.32 (Head of a representation). The dual notion of the socle is the head of a representation V ∈ Rep A; it is defined by \ def head V = all maximal subrepresentations of V Here, a subrepresentation M ⊆ V is called maximal if V /M is irreducible. The empty intersection is understood to be equal to V . Prove:

62

1 Representations of Algebras

(a) If V is finitely generated, then head V $ V . (b) If V is completely reducible, then head V = 0. Give an example showing that the converse need not hold. (c) If U ⊆ V is a subrepresentation such that V /U is completely reducible, then U ⊇ head V . If V is artinian (see Exercise 1.23), then the converse holds. (d) (rad A).V ⊆ head V ; equality holds if A is finite-dimensional. 1.33 (Matrix algebras). Let S ∈ Irr A. Viewing S ⊕n as a representation of Matn (A) as in Lemma 1.4(b), prove that S ⊕n is irreducible. 1.34 (Semisimple algebras). (a) Show that the algebra A is semisimple if and only if all short exact sequences in Rep A are split (Exercise 1.2). Qt (b) Let A ∼ = i=1 Matmi (Di ) be semisimple. Describe the ideals I of A and the factors A/I. Show that A has exactly t prime ideals and that there is a bijection ∼ Irr A ←→ Spec A, S ↔ Ker S. Moreover, all ideals I of A are idempotent: I 2 = I. 1.35 (Faithful completely reducible representations). Assume that A has a faithful completely reducible representation V . Show: (a) The algebra A is semiprime. In particular, if A is finite-dimensional, then A is semisimple. (b) If V is finite-dimensional, then A is finite-dimensional semisimple and Irr A is the set of distinct irreducible constituents of V . (c) The conclusion of (b) fails if V is not finite-dimensional: A need not be semisimple. 1.36 (Galois descent). Let K be a field, let Γ ≤ Aut(K), and assume that k = K Γ is the field of Γ -invariants in K. For a given k-vector space V , consider the action Γ K ⊗ V with γ ∈ Γ acting by γ ⊗ IdV ; so γ(λ ⊗ v) = γ(λ) ⊗ v. ∼ (a) Viewing V as a k-subspace of ResK k K ⊗V via V = k⊗V and the embedding Γ k ,→ K, show that V = (K ⊗ V ) = {x ∈ K ⊗ V | γ(x) = x for all γ ∈ Γ }, the space of Γ -invariants in K ⊗ V . (b) Let W ⊆ K ⊗ V be a Γ -stable K-subspace. Show that W = K ⊗ W Γ with Γ W = W ∩ V the space of Γ -invariants in W . 1.37 (Galois action). Let A be a k-algebra and let K/k be a finite Galois extension of fields with Galois group Γ = Gal(K/k). Show: (a) The kernel of any f ∈ HomAlgk (A, K) belongs to MaxSpec A. (b) For f, f 0 ∈ HomAlgk (A, K), we have Ker f = Ker f 0 if and only if f 0 = γ◦f for some γ ∈ Γ . 1.38 (Extension of scalars and complete reducibility). Let V be a representation of the k-algebra A. For a given field extension K/k, consider the representation K ⊗ V of K ⊗ A. Prove: (a) If K ⊗ V is completely reducible, then so is V . (b) If V is irreducible and K/k is finite separable, then K ⊗ V is completely reducible of finite length. Give an example showing that K ⊗ V need not be irreducible.

1.5 Characters

63

(c) If the field k is perfect and V is finite-dimensional completely reducible, then K ⊗ V is completely reducible for every field extension K/k. 1.39 (Extension of scalars and composition factors). Let V and W be finitedimensional representations of A and let K/k be a field extension. Prove: (a) If V and W are irreducible and non-equivalent, then the representations K⊗V and K ⊗ W of K ⊗ A have no common composition factor. (b) Conclude from (a) that, in general, K ⊗ V and K ⊗ W have a common composition factor if and only if V and W have a common composition factor. 1.40 (Splitting fields). Let A be a finite-dimensional k-algebra that is defined over some subfield k0 ⊆ k, in the sense that A ∼ = k ⊗k0 A0 for some k0 -algebra A0 . Assume that k is a splitting field for A. For a given field F with k0 ⊆ F ⊆ k, consider the F -algebra B = F ⊗k0 A0 . Show that F is a splitting field for B if and only if each S ∈ Irr A is defined over F , that is, S ∼ = k⊗F T for some representation T of B (necessarily irreducible). 1.41 (Dimension 2). Assume that k is perfect and let S ∈ Irr A with dimk S = 2. Show that if AS ⊆ Endk (S) is noncommutative, then D(S) = k. 1.42 (Separable algebras). An algebra A is called separable if there exists an eleP ment e = i xi ⊗ yi ∈ A ⊗ A satisfying m(e) = 1 and ae = ea for all a ∈ A. Here, m : A ⊗ A → A is the multiplication map and A ⊗ A is viewed as (A, A)-bimodule using left multiplicationP on the first factorPand right multiplication on the second; so P the conditions on e are: xi yi = 1 and i axi ⊗ yi = i xi ⊗ yi a. (a) Assuming A to be separable, show that K ⊗ A is separable for every field extension K/k. Conversely, if K ⊗A is separable for some K/k, then A is separable. (b) With e as above, show that the products xi yj generate A as k-vector space. Conclude that separable algebras are finite-dimensional. (c) Let V and W be representations of A. Regarding Homk (V, W ) as a representation of the algebra A ⊗ Aop as in Example 1.3 and viewing e ∈ A ⊗ Aop , show that e.f ∈ HomA (V, W ) for all f ∈ Homk (V, W ). Furthermore, assuming W to be a subrepresentation of V and f W = IdW , show that (e.f ) W = IdW . (d) Conclude from (a) – (c) that A is separable if and only if A is finitedimensional and K ⊗ A is semisimple for every field extension K/k. 1.43 (Centralizer subalgebras). Let A ⊆ B be semisimple k-algebras and let C = CB (A) denote the centralizer of A in B. Let W ∈ Irrfin B be absolutely irreducible and let V ∈ Irrfin A be such that H := HomA (V, ResB A W ) 6= 0. Let C act on H via c.φ = cW ◦ φ. Show that H ∈ Irr C.

1.5 Characters In this section, we will continue to focus on Repfin A, the full subcategory of Rep A consisting of the finite-dimensional representations of a k-algebra A. Analyzing a

64

1 Representations of Algebras

typical V ∈ Repfin A can be a rather daunting task, especially if the dimension of V is reasonably large. In this case, explicit computations with the operators aV (a ∈ A) involve expensive operations with large matrices. Fortunately, it often suffices to just know the traces of the operators aV ; these traces form the so-called character of V . For example, we shall see that if V is completely reducible and char k = 0, then V is determined up to isomorphism by its character (Theorem 1.42). Before proceeding, the reader may wish to have a quick look at Appendix B for the basics concerning traces of linear operators on a finite-dimensional vector space. Throughout this section, A denotes an arbitrary k-algebra. 1.5.1 Definition and Basic Properties The character of any V ∈ Repfin A is defined by

∈

k

∈

χV : A a

trace(aV )

(1.50)

Characters tend to be most useful if char k = 0; the following example gives a first illustration of this. Example 1.38 (The regular character). If the algebra A is finite-dimensional, then we can consider the character of the regular representation Areg , χreg = χAreg : A → k The regular character χreg is also denoted by TA/k when A or k need to be made explicit. For the matrix algebra A = Matn (k), one readily checks (Exercise 1.45) that χreg = n trace : Matn (k) → k In particular, χreg = 0 iff char k divides n. If K/k is a finite field extension, then we may view K as a finite-dimensional k-algebra. All finite-dimensional representations ⊕n for suitable n. It is a standard fact from field theory that of K are equivalent to Kreg χreg 6= 0 if and only if the extension K/k is separable (Exercise 1.48). Thus, by the lemma below, all characters χV of the k-algebra K vanish if K/k is not separable. Additivity The following lemma notes a basic property of characters: additivity on short exact sequences of representations. Lemma 1.39. If 0 → U → V → W → 0 is a short exact sequence in Repfin A , then χV = χU + χW .

1.5 Characters

65

∼ Proof. First note that if f : U −→ V is an isomorphism of finite-dimensional representations, then χV = χU . Indeed, aV = f ◦ aU ◦ f −1 holds for all a ∈ A by (1.21), and hence trace(aV ) = trace(aU ). f

g

Now let 0 → U −→ V −→ W → 0 be a short exact sequence of finitedimensional representations. Thus, Im f is an A-submodule of V such that U ∼ = Im f and W ∼ = V / Im f as A-modules. In view of the first paragraph, we may assume that U is an A-submodule of V and W = V /U . Choosing a k-basis of V that contains a k-basis of U , the matrix of each aV has block upper triangular form:   ∗ aU     0 aV /U Taking traces, we obtain trace(aV ) = trace(aU ) + trace(aV /U ) as desired.

t u

Multiplicativity Let V and W be representations of arbitrary algebras A and B, respectively; so we have algebra maps A → Endk (V ) and B → Endk (W ). By bi-functoriality of the tensor product of algebras (Exercise 1.10), we obtain an algebra map A ⊗ B → Endk (V ) ⊗ Endk (W ). The canonical map Endk (V ) ⊗ Endk (W ) → Endk (V ⊗ W ) in (B.16) is evidently also a map of algebras. Hence the composite gives the algebra map ∈

Endk (V ⊗ W )

∈

A⊗B a⊗b

aV ⊗ bW

(1.51)

making V ⊗ W a representation of A ⊗ B. This representation is called the outer tensor product of V and W ; it is sometimes written as V  W . If V and W are finite-dimensional, then (B.25) gives χV ⊗W (a ⊗ b) = χV (a)χW (b)

(1.52)

1.5.2 Spaces of Trace Forms Each character χV (V ∈ Repfin A) is a linear form on A – so χV ∈ A∗ – but more can be said: (i) By the standard trace identity trace(f ◦ g) = trace(g ◦ f ) for f, g ∈ Endk (V ), all characters vanish on the k-subspace [A, A] ⊆ A that is spanned by the Lie commutators [a, a0 ] = aa0 − a0 a in A. (ii) The character χV certainly also vanishes on Ker V ; note that this is a cofinite ideal of A.

66

1 Representations of Algebras

√ (iii) In fact, χV vanishes √ on the semiprime radical Ker V ; see Exercise 1.27. To see this, note that Ker V coincides with the√preimage of rad(A/ Ker V ) in A by Theorem 1.34 and so some power of √ Ker V is contained in Ker V by Proposition 1.23. Therefore, all elements of Ker V act as nilpotent endomorphisms on V . Since nilpotent endomorphisms have trace 0, it follows that √ Ker V ⊆ Ker χV . Below, we will formalize these observations. Universal trace and trace forms By (i), each character factors through the canonical map def

∈

Tr A = A/[A, A]

∈

Tr : A a

a + [A, A]

(1.53)

The map Tr satisfies the trace identity Tr(aa0 ) = Tr(a0 a) for a, a0 ∈ A; it will be referred to as the universal trace of A. Note that Tr gives a functor Algk → Vectk , because any k-algebra map φ : A → B satisfies f ([A, A]) ⊆ [B, B] and hence φ passes down to a k-linear homomorphism Tr φ : Tr A → Tr B. We will identify the linear dual (Tr A)∗ with the subspace of A∗ consisting of all linear forms on A that vanish on [A, A]. This subspace will be referred to as the space of trace forms on A and denoted by A∗trace Thus, all characters are trace forms. If φ : A → B is a k-algebra map, then the restriction of the dual φ∗ : B ∗ → A∗ in Vectk to the space of trace forms on B gives ∗ a map φ∗trace : Btrace → A∗trace . In this way, · ∗trace becomes a contravariant functor Algk → Vectk . Finite trace forms By (ii) and (iii), all characters belong to the following subspaces of A∗trace : A◦trace =

def



f ∈ A∗trace | f vanishes on some cofinite ideal of A

def



t ∈ A∗trace | t vanishes on some cofinite semiprime ideal of A

⊆



C(A) =



To see that these are indeed subspaces of A∗trace , observe that the intersection of any two cofinite (semiprime) ideals is again cofinite (semiprime). We will refer to A◦trace as the space of finite trace forms on A, because the space

1.5 Characters

67

A◦ = {f ∈ A∗ | f vanishes on some cofinite ideal of A } is generally called the finite dual of A; it will play a prominent role in Part IV of this book. As above, it is easy to see that · ◦ and · ◦trace are contravariant functors Algk → Vectk . To summarize, for any V ∈ Repfin A, we have χV ∈ C(A) ⊆ A◦trace ⊆ A∗trace ⊆ A∗

(1.54)

We will see shortly that, if k is a splitting field for A, then C(A) is the subspace of A∗ that is spanned by the characters of all finite-dimensional representations of A (Theorem 1.41). The case of finite-dimensional algebras Now assume that A is finite-dimensional. Then all ideals are cofinite and the Jacobson radical rad A is the unique smallest semiprime ideal of A (Proposition 1.23). Therefore, C(A) is exactly the subspace of A∗trace = A◦trace consisting of all trace ∗ forms that vanish on rad A. The latter space may be identified with (As.s )trace , where s.s A = A/ rad A is the semisimplification of A. So C(A) ∼ = A/[A, A] + rad A


∗

s.s ∗ s.s ∗ ∼ = (Tr A ) ∼ = (A )trace

(1.55)

Finite trace forms in positive characteristics We end our discussion of trace forms by giving another description of C(A) for a finite-dimensional algebra A over a field k with char k = p > 0. We also assume that k is a splitting field for A in the sense of §1.2.5. Put n def  T (A) = a ∈ A | ap ∈ [A, A] for some n ∈ Z+ One can show that T (A) is a k-subspace of A (Exercise 1.52). Proposition 1.40. Let A be finite-dimensional k-algebra and assume that k is a s.s ∼ ∼ splitting field
∗ for A and char k = p > 0. Then Tr A = A/T (A) and C(A) = A/T (A) . Proof. In light of the foregoing, it suffices to establish the first isomorphism. Since rad A is nilpotent by Proposition 1.23, we certainly have rad A ⊆ T (A). The canonical epimorphism A  As.s clearly maps T (A) into T (As.s ); in fact, one can show s.s that T (A) maps onto T (As.s ) (Exercise 1.52). Hence, T (A)/ rad A ∼ = T (A ). By s.s ∼ Q assumption of k, the Wedderburn decomposition A = S∈Irr Q A AS has compos.s nents AS ∼ = MatdS (k) with dS = dimk S. Clearly, T (A ) ∼ = S∈Irr A T (AS ), and we have also seen in the proof of Corollary 1.33 that the space [MatdS (k), MatdS (k)] consists of all trace-0 matrices and hence has codimension 1 in MatdS (k). Since the elementary idempotent matrix e1,1 does not belong to T (MatdS (k)), it fols.s ∼ Q lows that T (AS ) = [AS , AQ ]. Therefore, T (A ) = S S∈Irr A [AS , AS ] and hence s.s s.s s.s A/T (A) ∼ t u = A /T (A ) ∼ = S∈Irr A Tr AS ∼ = Tr A . This finishes the proof.

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1.5.3 Irreducible Characters Characters of finite-dimensional irreducible representations are referred to as irreducible characters. Since every finite-dimensional representation has a composition series, all characters are sums of irreducible characters by Lemma 1.39. The following theorem lists some important properties of the collection of irreducible characters of an arbitrary algebra A. For a minor extension of part (a), see Exercise 1.50. Recall that the base field k of A is said to be a splitting field if D(S) = k holds for all S ∈ Irrfin A. Theorem 1.41. Let A be a k-algebra. Then: (a) The irreducible characters χS for S ∈ Irrfin A such that char k does not divide dimk D(S) are linearly independent. (b) # Irrfin A ≤ dimk C(A) (c) If k is a splitting field, then the irreducible characters of A form a k-basis of C(A). In particular, # Irrfin A = dimk C(A) holds in this case. Pr Proof. (a) Suppose that i=1 λi χSi = 0 for suitable λi ∈ k and Si ∈ Irrfin A such that char k does not divide dimk D(Si ). We need to show that all λi = 0. By Theorem 1.34, the annihilators Ker Si are distinct maximal ideals of A, and A/ Ker Si ∼ = Bi := Matmi (Di ) by Burnside’s Theorem (§1.4.5). The Chinese Remainder Theorem yields an epimorphism of algebras r Y Tr A A/ i=1 Ker Si ∼ B := Bi ∈

∈

i=1

a

aSi

(1.56)




Let ei ∈ B be the element corresponding to the t-tuple with the elementary matrix e1,1 ∈ Matmi (Di ) in the ith component and the 0-matrix in all other components. It is easy to see (Exercise 1.45(a)) that χSi (ej ) = dimk (Di )1k δi,j , because ⊕mi A acts Pon Si via the standard action of Matmi (Di ) on Di . We conclude that 0 = i λi χSi (ej ) = λj dimk (Dj )1k for all j. Finally, our hypothesis implies that op dimk (Dj )1k 6= 0, since Dj ∼ = D(Sj ) , giving the desired conclusion λj = 0. (b) Let S1 , . . . , Sr be nonequivalent finite-dimensional irreducible representations of A and consider the epimorphism (1.56). Since B is finite-dimensional L ∗ semiprime, we have Tr(B)∗ ,→ C(A). Moreover, clearly, Tr(B)∗ ∼ = i Tr(Bi ) . Thus, it suffices to show that every finite-dimensional k-algebra B has a nonzero e = B ⊗ k, where k is an algebraic trace form. To see this, consider the algebra B e e closure of k, and fix some S ∈ Irr B. The character χSe is nonzero by (a) and its e as restriction to B is nonzero as well, because the canonical image of B generates B k-vector space. Composing χSe with a suitable k-linear projection of k onto k yields the desired trace form. (c) Since the collection of all irreducible characters is linearly independent by (a), it suffices to show that they span C(A). But any t ∈ C(A) vanishes some cofinite semiprime ideal I of A. The algebra A/I is split semisimple by Theorem 1.35

1.5 Characters

69

and the trace form t can be viewed as an element of C(A/I) = Tr(A/I)∗ . Since # Irr A/I = dimk Tr(A/I)∗ by Corollary 1.33(a), we know that t is a linear combination of irreducible characters of A/I. Viewing these characters as (irreducible) characters of A by inflation, we have written t as a linear combination of irreducible characters of A. This finishes the proof. t u Characters of completely reducible representations It is a fundamental fact of representation theory that, under some restrictions on char k, finite-dimensional completely reducible representations are determined up to equivalence by their character: Theorem 1.42. Let V and W be finite-dimensional completely reducible representations of a k-algebra A such that char k = 0 or char k > max{dimk V, dimk W }. Then V ∼ = W if and only if χV = χW . Proof. Since V ∼ = W clearly always implies χV = χW (Lemma 1.39), let us assume that χV = χW and V ∼ on char k. To this = W under the L given hypotheses Lprove that⊕m ⊕nS S ∼ ∼ with mS = and W S end, write V = S = S∈Irr S∈Irr A P A Pm(S, V ) and nS = m(S, W ). Lemma 1.39 gives χV = S mS χS and χW = S nS χS , and we need to show that mS = nS for all S. But mS 6= 0 or nS 6= 0 implies that S is finite-dimensional, with dimk S bounded above by dimk V or dimk W , and our hypothesis on the characteristic also implies that char k does not divide dimk D(S), because dimk D(S) is a divisor of dimk S.P Therefore, Theorem 1.41(a) allows us to deduce from the equality 0 = χV − χW = S (mS − nS )χS that (mS − nS )1k = 0 if mS 6= 0 or nS 6= 0. Since |mS − nS | ≤ max{dimk V, dimk W } in this case, our hypothesis on char k implies that mS = nS , thereby finishing the proof. t u 1.5.4 The Grothendieck Group R(A) Certain aspects of the representation theory of a given k-algebra A discussed in this chapter, especially those related to characters, can be conveniently packaged with the aid of the Grothendieck group of finite-dimensional representations of A. This group will be denoted by R(A) By definition, R(A) is the abelian group with generators [V ] for V in Repfin A and with relations [V ] = [U ] + [W ] for each short exact sequence 0 → U → V → W → 0 in Repfin A. Formally, R(A) is the factor of the free abelian group on the set of all isomorphism classes (V ) of finite-dimensional representations V of A – these isomorphism classes do indeed form a set – modulo the subgroup that is generated by the elements (V )−(U )−(W ) for each short exact sequence 0 → U → V → W → 0 of finite-dimensional representation of A. The generator [V ] is the image of (V ) in R(A). The point of this

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construction is as follows. Suppose we have a rule that assigns to every V ∈ Repfin A a value f (V ) ∈ G, where G is some abelian group, in such a way that the assigment is additive on short exact sequences 0 → U → V → W → 0 in Repfin A in the sense that f (V ) = f (U ) + f (W ) holds for each such sequence. Then we obtain a well-defined group homomorphism f : R(A) → G ,

[V ] 7→ f (V )

We remark that other notations for R(A) are also used in the literature, for example R(A) is denoted by Gk0 (A) in Swan [171]. Group theoretical structure Note that if 0 = V0 ⊆ V1 ⊆ · · · ⊆ Vl = V is any chain of finite-dimensional representations, then the relations of R(A) (and a straightforward induction) imply that l X [Vi /Vi−1 ] [V ] = i=1

In particular, taking a composition series for V , we see that R(A) is generated by the elements [S] with S ∈ Irrfin A. In fact, these generators are independent: Proposition 1.43. The group R(A) is isomorphic to the free abelian group generated by the set Irrfin A of isomorphism classes of finite-dimensional irreducible representations of A. An explicit isomorphism is given by multiplicities,

∈

∈

µ : R(A) ∼ Z⊕ Irrfin A [V ]


µ(S, V ) S

Proof. The map µ yields a well-defined group homomorphism by virtue of the fact that multiplicities are additive on short exact sequences by (1.31). For S ∈ Irrfin A,
one has µ([S]) = δS,T T . These elements form the standard Z-basis of Z⊕ Irrfin A . Therefore, the generators [S] are Z-independent and m is an isomorphism. t u Functoriality Pulling back representations along a given algebra map φ : A → B as in §1.2.2 turns short exact sequences in Repfin B into short exact sequences in Repfin A. Therefore, we obtain a group homomorphism

∈

R(A)

∈

R(φ) : R(B) [V ]

[φ∗ V ]

In this way, we may view R as a contravariant functor from Algk to the category AbGroups ≡ Z Mod of all abelian groups.

1.5 Characters

71

Lemma 1.44. Let φ : A  B be an epimorphism of algebras. Then R(φ) is a split injection coming from the inclusion φ∗ : Irrfin B ,→ Irrfin A : R(B)

R(φ)

R(A) ∼

∼ Z⊕ Irrfin B

Z⊕ Irrfin A

If Ker φ ⊆ rad A then R(φ) is an isomorphism. Proof. The first assertion is immediate from Proposition 1.43, since inflation φ∗ clearly gives inclusions Irr B ,→ Irr A and Irrfin B ,→ Irrfin A. If Ker φ ⊆ rad A then these inclusions are in fact bijections by (1.41). t u Extension of the base field Let K/k be a field extension. For any A ∈ Algk and any V ∈ Repfin A, consider algebra K ⊗ A ∈ AlgK and the representation K ⊗ V ∈ Repfin (K ⊗ A) as in (1.24). By exactness of the scalar extension functor K ⊗ · , this process leads to a well-defined group homomorphism

∈

R(K ⊗ A)

∈

K ⊗ · : R(A) [V ]

[K ⊗ V ]

(1.57)

Lemma 1.45. The scalar extension map (1.57) is injective. Proof. In view of Proposition 1.43, it suffices to show that, for S 6= T ∈ Irrfin A, the representations K ⊗ S, K ⊗ T ∈ Repfin (K ⊗ A) have no common composition factor. To prove this, we may replace A by A/ Ker S ∩ Ker T , thereby reducing to the case where A is semisimple. The central primitive idempotent e(S) ∈ A acts as the identity on S and as 0 on W ; see (1.47). Viewed as an element of K ⊗ A, we have the same actions of e(S) on K ⊗S and on K ⊗T , whence these representations cannot have a common composition factor. t u The character map Since characters are additive on short exact sequences in Repfin A by Lemma 1.39, they yield a well-defined group homomorphism

∈

C(A)

∈

χ : R(A) [V ]

χV

A◦trace (1.58)

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1 Representations of Algebras

where C(A) is the subspace of A◦trace consisting of all trace forms that vanish on some cofinite semiprime ideal of A as in (1.54). The character map is natural in A, that is, for any morphism φ : A → B in Algk , it is clear the following diagram commutes: χ

R(B)

◦ Btrace

(1.59)

◦

R(φ)

φtrace χ

R(A)

A◦trace

Next, we consider the k-vecorspace def

Rk (A) = R(A) ⊗Z k Since C(A) is a k-vector space, χ lifts uniquely to a k-linear map χk : Rk (A) −→ C(A) Proposition 1.46. The map χk is injective. If k is a splitting field for A, then χk is an isomorphism. Proof. First assume that k is a splitting field for A. By Proposition 1.43, the classes [S] ⊗ 1 with S ∈ Irrfin A form a k-basis of Rk (A), and by Theorem 1.41, the irreducible characters χS form a k-basis of C(A). Since χk ([S] ⊗ 1) = χS , the proposition follows in this case. If k is arbitrary, the fix an algebraic closure k and consider the algebra A = k⊗A. Every trace form A → k extends uniquely to a trace form A → k, giving a map ∗ A∗trace → Atrace (which is in fact an embedding). The following diagram is evidently commutative: Rk (A)

χk

A∗trace (1.60)

k⊗ ·

Rk (A)

∗

χk

Atrace

Here, k ⊗ · and χk are injective by Lemma 1.45 and the first paragraph of this proof, respectively, whence χk must be injective as well. t u Positive structure and dimension augmentation The set

def

R(A)+ =



[V ] | V ∈ Repfin A



is a submonoid of the group R(A), because 0 = [0] ∈ R(A)+ and [V ] + [V 0 ] = [V ⊕ V 0 ] ∈ R+ (A) for M, M 0 in Repfin A; it is called the positive cone of R(A).

1.5 Characters

73

⊕ Irrfin A Under the isomorphism R(A) ∼ of Proposition 1.43, the positive cone = Z ⊕ Irrfin A R(A)+ corresponds to Z+ . In particular, R(A) = R(A)+ − R(A)+ – so every element of R(A) is a difference of two elements of R(A)+ – and 0 is the only x ∈ R(A) for which both x ∈ R(A)+ and −x ∈ R(A)+ . This also follows from the fact that R(A) is equipped with a group homomorphism

∈

(Z, +)

∈

dim : R(A) [V ]

dimk V

called the dimension augmentation of R(A), and dim x > 0 for 0 6= x ∈ R(A)+ . Defining def x ≤ y ⇐⇒ y − x ∈ R(A)+ we obtain a translation invariant partial order on the group R(A).

Exercises Unless specified otherwise, A denotes an arbitrary k-algebra in these exercises. 1.44 (Idempotents). Let e = e2 ∈ A be an idempotent and let V be in Rep A. Show: (a) For every subrepresentation U ⊆ V , one has U ∩ e.V = e.U . (b) If V is finite-dimensional, then χV (e) = (dimk e.V )1k 1.45 (Matrices). (a) Let V be a finite-dimensional representation of A. Viewing ⊕n V P as a representation of Matn (A) as in Lemma 1.4(b), show that χV ⊕n (a) = i χV (ai,i ) for a = (ai,j ) ∈ Matn (A). (b) For the matrix algebra Matn (k), show that all characters are of the form k trace with k ∈ Z+ and that χreg = n trace. (c) Let Tr : A  A/[A, A] be
the P universal trace as in (1.53). Show that the map Matn (A) → A/[A, A], ai,j 7→ i Tr(ai,i ), yields a k-linear isomorphism Matn (A)/[Matn (A), Matn (A)] ∼ = A/[A, A]. 1.46 (The regular character for different base fields). Let A be a finite-dimensional k-algebra and let K a subfield of A with k ⊆ K ⊆ Z (A). Thus, we may view A as a (finite-dimensional) K-algebra. Let TA/k and TA/K denote the regular characters of A when viewed as a k-algebra and as a K-algebra, resp., and let TK/k denote the regular character of K as a k-algebra. Show that TA/k = TK/k ◦ TA/K . 1.47 (Finite-dimensional central simple algebras). This exercise uses definitions and results from Exercise 1.13. Let A be a finite-dimensional k-algebra. (a) Show that finite-dimensional central simple algebras are separable in the ∼ sense of Exercise 1.42. More precisely, show that A is central √ simple iff k ⊗ A = Matn (k), where k denotes an algebraic closure of k and n = dimk A. (b) If A is central simple, show that the regular character χreg = TA/k vanishes if and only if char k divides dimk A.

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1 Representations of Algebras

1.48 (Finite separable field extensions). Let K/k be a finite field extension. Recall from field theory (e.g., [109, V.4 and V.6]) that K/k is a separable field extension iff there are [K : k] distinct k-algebra embeddings σi : K ,→ k, where k is an algebraic closure of k. If K/k is not separable, then char k = p > 0 and there is an intermediate field k ⊆ F $ K such that the minimal polynomial of any α ∈ K over r F has the form xp − a for some r ∈ N and a ∈ F . Show that the following are equivalent: (i) K/k is separable; [K:k] (ii) K ⊗ k ∼ as k-algebras; =k (iii) K is a separable k-algebra in the sense of Exercise 1.42; (iv) the regular character TK/k (Exercise 1.46) is nonzero. 1.49 (Separable algebras, again). (a) Show that a k-algebra A is separable in the sense of Exercise 1.42 if and only if A is finite-dimensional semisimple and Z (D(S))/k is a separable field extensions for all S ∈ Irr A. (b) Let A be a finite-dimensional k-algebra that is defined over a perfect field k0 ⊆ k as in Exercise 1.40: A = k ⊗k0 A0 for some k0 -algebra A0 ⊆ A. Show that rad A = k ⊗k0 rad A0 . 1.50 (Nonzero irreducible characters). For S ∈ Irrfin A, put K(S) = Z (D(S)). (a) Show that χS 6= 0 if and only if char k does not divide dimK(S) D(S) and K(S)/k is separable. (b) Show that the nonzero irreducible characters χS are k-linearly independent. 1.51 (Algebras that are defined over finite fields). Let A be a finite-dimensional kalgebra such that A ∼ = k ⊗k0 A0 for some finite subfield k0 ⊆ k and some k0 -algebra A0 . (a) Let S ∈ Irr A be absolutely irreducible and let F = F (S) be the subfield of k that is generated by k0 and the character values χS (a) with a ∈ A0 . Show that F is finite and that S ∼ = k ⊗F T for some T ∈ Irr B, where B = B(S) = F ⊗k0 A0 . [Use Wedderburn’s Theorem on finite division algebras.] (b) Assume that k is a splitting field for A and let F be the (finite) subfield of k that is generated by k0 and the character values χS (a) with a ∈ A0 and S ∈ Irr A. Show that F is a splitting field for B = F ⊗k0 A0 . [Use Exercise 1.40.] 1.52 (The Frobenius endomorphism of Tr(A)). Let A be an arbitrary algebra over a field k with char k = p > 0 and let n  T (A) = a ∈ A | ap ∈ [A, A] for some n ∈ Z+ as in Proposition 1.40. (a) Show that (a + b)p ≡ ap + bp mod [A, A] for all a, b ∈ A. Furthermore, if a ∈ [A, A] then ap ∈ [A, A]. Conclude that the pth -power map yields a well-defined group endomorphism of Tr(A) = A/[A, A]. (b) Conclude from (a) that T (A) is a k-subspace of A. (c) Let I be a nilpotent ideal of A. Show that T (A) maps onto T (A/I) under the canonical epimorphism A  A/I.

1.5 Characters

75

1.53 (Irreducible representations of tensor products). (a) Let S and T be finitedimensional absolutely irreducible representations of algebras A and B, respectively. Use Burnside’s Theorem (§1.4.5) to show that the outer tensor product S  T is an absolutely irreducible representation of A ⊗ B; see (1.51). (b) Assuming A and B to be split semisimple, show that A⊗B is split semisimple as well and all its irreducible representation arise as in (a). 1.54 (More on Grothendieck groups). Let A be a k-algebra. (a) (b) General G0 (A) plus ?? (c) Functoriality: induction. 1.54 (a)

2 Further Topics on Algebras and their Representations

At a first pass through this book, the material in this chapter can be skipped and referred to as the need arises.

2.1 Projectives So far, the focus has been on irreducible and completely reducible representations. For algebras A that are not necessarily semisimple, another class of representations also plays an important role, the projective modules of A. We will also refer to them as the projectives of A for short; we shall however refrain from calling projective modules “projective representations”, since this term is reserved for group homomorphisms of the form G → PGL(V ), where V is a vector space. 2.1.1 Definition and Basic Properties Let A be an arbitrary algebra. A module P ∈ A Mod is called projective if P is a di0 rect summand of some free module: P 0 ⊕ Q = A⊕I reg for suitable P , Q ∈ A Mod with P0 ∼ = P and some set I. Projective modules, like free modules, can be thought of as approximate “vector spaces over A”, but projectives provide a much more ample and stable class of modules than free modules. Proposition 2.1. The following are equivalent for a module P ∈ A Mod. (i) P is projective; (ii) Given an epimorphism f : M  N and an arbitrary p : P → N in A Mod, there exists a “lift” pe: P → M in A Mod such that f ◦ pe = p. P ∃p e

M

f

p

N

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2 Further Topics on Algebras and their Representations

(iii) Every epimorphism f : M  P in A Mod splits: there exists a homomorphism s : P → M in A Mod such that f ◦ s = IdP . 0 ∼ Proof. (i) =⇒ (ii): Say P 0 ⊕ Q = F with F = A⊕I reg and P = P . Identifying P 0 with P , as we may, the embedding µ : P ,→ F and the projection π : F  P along Q satisfy π ◦ µ = IdP

Consider the module map q = p ◦ π: F → N In order to construct a lift for q, fix an A-basis {fi }i∈I for F . Since the map f is surjective, there are {mi }i∈I ⊆ M such that f (mi ) = q(fi ) for all i. Now define the desired lift qe: F → M on the chosen basis by qe(fi ) = mi ; this determines qe unambiguously on F and we have f ◦ qe = q Putting pe = qe ◦ µ : P → M , we obtain f ◦ pe = f ◦ qe ◦ µ = q ◦ µ = p ◦ π ◦ µ = p ◦ IdP = p as desired. (ii) =⇒ (iii): Taking N = P and p = IdP , the lift pe: P → M from (ii) will serve as the desired splitting map s. (iii) =⇒ (i): As we have observed before, any generating set {xi | i ∈ I} of P P gives rise to an epimorphism f : F = A⊕I reg  P , (ai ) 7→ i ai xi . If s is the 0 0 ∼ splitting provided by (iii), then F = P ⊕ Q with P = Im s = P and Q = Ker f (Exercise 1.2), proving (i). t u Functorial aspects Any M ∈ A Mod gives rise to a functor HomA (M, · ) : A Mod −→ Vectk with HomA (M, f ) = f ◦ · : HomA (M, X) → HomA (M, Y ) for a morphism f : X → Y in A Mod. We will write f∗ = HomA (M, f ). f

g

The functor HomA (M, · ) is left exact: if 0 → X −→ Y −→ Z → 0 is a f∗ short exact sequence in A Mod, then the resulting sequence 0 → HomA (M, X) −→ g∗ HomA (M, Y ) −→ HomA (M, Z) is exact in Vectk , that is, f∗ is mono and Im f∗ = Ker g∗ (§B.2.1). However, g∗ need not be epi. In fact, calling a functor F : A Mod −→ Vectk exact if it turns short exact sequences into short exact sequences, characterization (ii) in Proposition 2.1 can be reformulated as follows: M ∈ A Mod is projective if and only if HomA (M, · ) is exact.

(2.1)

2.1 Projectives

79

In lieu of HomA (M, · ), we could of course equally well consider the (contravariant) functor HomA ( · , M ) : A Mod −→ Vectk and wonder when it is exact. The A-modules M for which this holds are called injective; they are explored in Exercise 2.3. The Dual Bases Lemma We now turn to a more elementwise characterization of projectivity. Again, let A be an arbitrary k-algebra and let M ∈ A Mod. Put def

M ∨ = HomA (M, Areg ) and let h . , . i : M ∨ × M → A denote the evaluation pairing. A pair of indexed sets {xi }i∈I ⊆ M and {xi }i∈I ⊆ M ∨ is called a pair of dual bases for M if, for each x ∈ M, • •

hxi , xi P= 0 for almost all i ∈ I, and x = i∈I hxi , xi.xi .

We can equip M ∨ with a right A-module structure by defining hf a, mi = hf, mia for f ∈ M ∨ , a ∈ A and m ∈ M . With this, we have the following canonical homomorphism in Vectk :

∈

EndA (M )

∈

M ∨ ⊗A M f ⊗m


x 7→ hf, xim

(2.2)

It is straightforward to check that the image of this map is an ideal of EndA (M ) (Exercise 2.4). For A = k, part (b) of the following lemma reduces to the standard ∗ ∗ isomorphism Endk (V ) ∼ = V ⊗V ∼ = V ⊗V for a finite-dimensional k-vector space V ; see Appendix B. Lemma 2.2. Let M ∈ A Mod. (a) A pair of dual bases {(xi , xi )}i∈I ⊆ M × M ∨ exists if and only if M is projective. The families {xi }i∈I which can occur in this case are exactly the generating families of M . (b) The map (2.2) is an isomorphism if and only if M is finitely generated projective. Proof. (a) For given generators {xi }i∈I ⊆ M , consider the epimorphism A⊕I reg  P M , (ai )i 7→ i ai xi . If M is projective, then we may fix a splitting s : M → A⊕I reg . ⊕I th Now let πi : Areg → A, (ai )i 7→ ai , be the projection onto the i component and define xi = πi ◦ s ∈ HomA (M, Areg ) to obtain the desired pair of dual bases. Conversely, if {(xi , xi )}i∈I are dual bases for M , then {xi }i∈I ⊆ M generates i ⊕I M and the map M → A⊕I reg , x 7→ (hx , xi)i , splits the epimorphism Areg  M ,

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2 Further Topics on Algebras and their Representations

P ⊕I (ai )i 7→ i ai xi . Therefore, M is isomorphic to a direct summand of Areg , and hence M is projective. Pn (b) Let µ denote the map (2.2) and note that µ( i=1 fi ⊗ mi ) = IdM says n exactly that {(fi , mi )}1 are dual bases for M . Hence we know by (a) that IdM ∈ Im µ if and only if M is finitely generated projective. Furthermore, since Im µ is an ideal of EndA (M ), the condition IdM ∈ Im µ is equivalent to surjectivity of µ. Consequently, µ is surjective if and only if M is finitely generated projective. It remains to remark that µ is also injective in this case. Indeed, by embedding M as ⊕n a direct summand into some A⊕n reg , one reduces to case M = Areg , which in turn follows from Lemma 1.5(b). We leave it to the reader to work out the details of the argument (Exercise 2.4). t u Categories of projectives For any algebra A, let A Proj

,

A proj

and

Projfin A

denote the full subcategories of A Mod consisting of all projectives, the finitely generated projectives, and the finite-dimensional projectives of A, respectively. We will be primarily concerned with the latter two. Induction along an algebra homomorphism α : A → B gives a functor α∗ = IndB A : A Proj → B Proj B For, if P is a direct summand of A⊕I reg for some set I, then IndA P = B ⊗A P is a ⊕I ⊕I B direct summand of Breg = IndB A Areg , because IndA commutes with direct sums. If P is finitely generated, then I may be chosen finite. Thus, induction restricts to a functor IndB A : A proj → B proj

Any P ∈ A proj can be concretely realized by means of idempotent matrices over µ A. Indeed, there are A-module maps P A⊕n reg for some n with π ◦ µ = IdP . op π ⊕n ∼ Then µ ◦ π ∈ EndA (Areg ) = Matn (A) (Lemma 1.5) gives an idempotent matrix e with ⊕n (2.3) P ∼ = Areg · e In terms of matrices, induction can be desribed as follows: ∼ ⊕n IndB A P = Breg · Matn (α)(e) 2.1.2 Hattori-Stallings Traces For P ∈ A proj, the Dual Basis Lemma (Lemma 2.2) allows us to consider the following map which was introduced independently by Hattori [82] and Stallings [169]:

2.1 Projectives

∈

Tr A = A/[A, A]

∈

∨ TrP : EndA (P ) ∼ = P ⊗A P

81

f ⊗p

hf, pi + [A, A]

(2.4)

Observe that this map is k-linear in both f and p and hf a, pi ≡ hf, api mod [A, A] for a ∈ A. Therefore, TrP is a well-defined k-linear map. For A = k, we recover the standard trace map (B.22) and the following rank of P equals dimk P : def

rank P = TrP (IdP ) ∈ Tr A

(2.5)

Using dual bases {(xi , xi )}ni=1 ⊆ P × P ∨ for P , the rank of P may be expressed as X i rank P = hx , xi i + [A, A] (2.6) i ⊕n 2 Alternatively, writing P ∼ = Areg P· e for some idempotent matrix e = e ∈ Matn (A) as in (2.3), one has rank P = i eii + [A, A] (Exercise 2.10). In particular,

rank A⊕n reg = n + [A, A] The following lemma summarizes some essential properties of the HattoriStallings trace. Lemma 2.3. Let A be a k-algebra and let P, P 0 ∈ A proj. (a) TrP (φ ◦ ψ) = TrP (ψ ◦ φ) for all φ, ψ ∈ EndA (P ). (b) If φ ∈ EndA (P ), φ0 ∈ EndA (P 0 ), then TrP ⊕P 0 (φ ⊕ φ0 ) = TrP (φ) + TrP 0 (φ0 ). In particular, rank(P ⊕ P 0 ) = rank P + rank P 0 . ∨ Proof. (a) Viewing the isomorphism EndA (P ) ∼ = P ⊗A P as an identification, it suffices to establish the trace property TrP (φ ◦ ψ) = TrP (ψ ◦ φ) for φ = f ⊗ p and ψ = f 0 ⊗ p0 . Note that composition in EndA (P ) takes the form

φ ◦ ψ = (f ⊗ p) ◦ (f 0 ⊗ p0 ) = hf, p0 if 0 ⊗ p Therefore, TrP (φ ◦ ψ) = hf, p0 ihf 0 , pi + [A, A] = hf 0 , pihf, p0 i + [A, A] = TrP (ψ ◦ φ). (b) Put Q = P ⊕ P 0 and Φ = φ ⊕ 0P 0 , Φ0 = 0P ⊕ φ0 ∈ EndA (Q). Then φ ⊕ φ = Φ + Φ0 . It is easy to see that TrQ (Φ) = TrP (φ) and similarly for Φ0 (Exercise 2.7). Thus, the desired formula TrP ⊕P 0 (φ ⊕ φ0 ) = TrP (φ) + TrP 0 (φ0 ) follows by linearity of TrQ . Since IdQ = IdP ⊕ IdP 0 , we also obtain the rank formula rank Q = rank P + rank P 0 . t u

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2 Further Topics on Algebras and their Representations

Functoriality Lastly, we briefly address the issue of changing the algebra. So let α : A → B be an algebra map. The we have the k-linear map Tr α : Tr A → Tr B be as in §1.5.2. Furthermore, for any P ∈ A proj and any φ ∈ EndA (P ), we may consider the induced module α∗ P = IndB A P ∈ B proj and the endomomorphism α∗ φ = IndB A φ ∈ EndB (α∗ P ). We leave it to the reader (Exercise 2.7) to verify the formula Trα∗ P (α∗ φ) = (Tr α)(TrP (φ)) (2.7) With φ = IdP , this gives in particular rank(α∗ P ) = (Tr α)(rank P )

(2.8)

⊕n With matrices, this can also be seen as follows. Write P ∼ = Areg · e as in (2.3). Then P ∼ ⊕n rank P P = i eii + [A, A] and α∗ P = B · Matn (α)(e), and so rank(α∗ P ) = i α(eii ) + [B, B] = (Tr α)(rank P ).

2.1.3 The Grothendieck Groups K0 (A) and P(A) We continue to let A denote an arbitrary algebra. The Grothendieck group of finitely generated projectives Using A proj in place of Repfin A, one can construct an abelian group along the exact same lines as the construction of R(A) in §1.5.4: The group has generators [P ] for P ∈ A proj and relations [Q] = [P ] + [R] for each short exact sequence 0 → P → Q → R → 0 in A proj. Since all these sequences split by Proposition 2.1, this means that we have a relation [Q] = [P ]+[R] whenever, Q ∼ = P ⊕R in A proj. The resulting abelian group is commonly denoted by K0 (A) For P, Q ∈ A proj, the equality [P ] = [Q] in K0 (A) means that P and Q are stably ⊕r ∼ isomorphic in the sense that P ⊕ A⊕r reg = Q ⊕ Areg for some r ∈ Z+ (Exercise 2.9). The construction of K0 is functorial. Indeed, if α : A → B is an algebra homomorphism, then we have already seen that induction yields a functor IndB A : A proj → B proj that commutes with direct sums. Therefore, we have a well-defined homomorphism of abelian groups,

∈

K0 (B)

∈

K0 (α) : K0 (A) [P ]

[IndB A P]

In this way, we obtain a functor K0 : Algk → AbGroups.

2.1 Projectives

83

We remark that, in principle, it would of course be possible to perform analogous constructions using the full subcategory of A Mod consisting of all projectives of A, not just the finitely generated ones. However, the resulting group would be trivial in this case; see Exercise 2.11. Finite-dimensional projectives For the purposes of representation theory, we shall often be concerned with a smaller category of projectives than A proj, namely the full subcategory of A proj consisting of all finite-dimensional projectives of A. This category will be denoted by Projfin A and the corresponding Grothendieck group, constructed from Projfin A as above, will be denoted by P(A) Part (a) of the following proposition sorts out the group theoretical structure of P(A) in a manner analogous to Proposition 1.43 which did the same for P(A). While the latter result was a consequence of the Jordan-Hölder Theorem (Theorem 1.18), the operative fact in Proposition 2.4 below is the Krull-Schmidt Theorem (§1.2.6). Proposition 2.4. (a) The group P(A) is isomorphic to the free abelian group generated by the set of isomorphism classes of finite-dimensional indecomposable projectives of A. (b) For P, Q ∈ Projfin A , the equality [P ] = [Q] in P(A) is equivalent to P ∼ = Q. Proof. By the Krull-Schmidt Theorem, any P ∈ Projfin A can be decomposed into a finite direct sum of indecomposable summands and this decomposition is unique up to the order the summands and their isomorphism type. Thus, letting I denote a set of representatives for isomorphism classes of finite-dimensional indecomposable projectives of A, we have M ⊕n (P ) I I P ∼ = I∈I

for unique nI (P ) ∈ Z+ almost all of which are zero. Evidently, nI (P ⊕ Q) = nI (P ) + nI (Q) for all P, Q ∈ Projfin A, and so we obtain a well-defined group homomorphism ∈

Z⊕I

∈

P(A) [P ]

nI (P )


I

The map sending the standard Z-basis element eI = (δI,J )J ∈ Z⊕I to [I] ∈ P(A) is inverse to the above homomorphism, and so we have in fact constructed an isomorphism. This proves (a), and (b) is an immediate consequence as well. t u

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2 Further Topics on Algebras and their Representations

The Cartan homomorphism Since any P ∈ Projfin A is of course also an object of A proj and of Repfin A, the symbol [P ] can be interpreted in P(A) as well as in K0 (A) and in R(A). In fact, it is clear from our definition of these groups that there are group homomorphisms

∈

R(A)

∈

c : P(A) [P ]

[P ]

(2.9)

and an analogous homomorphism P(A) → K0 (A). The map (2.9) is particularly important; it is called the Cartan homomorphism. Despite the deceptively simple looking expression above, c need not be injective (Exercise 2.14) whereas the homomorphism P(A) → K0 (A) is in fact always an embedding (Exercise 2.9). A pairing between K0 (A) and R(A) Let P ∈ A proj and V ∈ Repfin A. Then the k-vector space HomA (P, V ) is finite⊕n ∼ dimensional, because A⊕n reg  P for some n and so HomA (P, V ) ,→ HomA (Areg , V ) = ⊕n V . Thus, we may define def

(P, V ) = dimk HomA (P, V ) Since the functor HomA (P, · ) is exact by (2.1), we obtain a group homomorphism (P, · ) : R(A) → Z, and since the functor HomA ( · , V ) does at least respect direct sums, we also have a group homomorphism ( · , V ) : P(A) → Z. Thus, (P, V ) only depends on the classes [P ] ∈ P(A) and [V ] ∈ R(A) and we have a bi-additive pairing ( · , · ) : K0 (A) × R(A) ∈

∈

Z

([P ], [V ])

(P, V )

(2.10)

Under suitable hypotheses, this pairing will give a “duality” between K0 (A) and R(A) that will play an important role in subsequent chapters. Hattori-Stallings ranks and characters By Lemma 2.3(b), the Hattori-Stallings rank gives a well-defined group homomorphism ∈

Tr A

∈

rank : K0 (A) [P ]

rank P

(2.11)

2.1 Projectives

85

If α : A → B is a homomorphism of k-algebras, then the following diagram commutes by (2.8): K0 (A)

K0 (α)

K0 (B)

rank

(2.12)

rank

Tr A

Tr α

Tr B

The following proposition, due to Bass [9], further hints at the aforementioned duality between K0 (A) and R(A) by relating the rank map (2.11) to the character homomorphism χ : R(A) → A∗trace in (1.58). Proposition 2.5. For any algebra A, the following diagram commutes: K0 (A) × R(A)

(·, ·)

Z

rank ×χ

can.

Tr A × A∗trace

evaluation

k

Proof. The proposition states that, for P ∈ A proj and V ∈ Repfin A, hχV , rank P i = dimk HomA (P, V ) 1k ⊕n ∼ V ⊕n and hχV , rank P i = This is clear if P ∼ = Areg ; for, then HomA (P, V ) = hχV , n + [A, A]i = n dimk V . µThe general case elaborates on this observation. In F = A⊕n detail, fix A-module maps P reg for some n with π ◦ µ = IdP . The π functor HomA ( · , V ) then yields k-linear maps ∗

HomA (P, V )

π = · ◦π ∗

µ = · ◦µ

⊕n HomA (F, V ) ∼ =V

with µ∗ ◦ π ∗ = IdHomA (P,V ) . Thus, h := π ∗ ◦ µ∗ ∈ Endk (HomA (F, V )) is an idempotent with Im h ∼ = HomA (P, V ). Therefore, by standard linear algebra (Exercise 1.44(b)), dimk HomA (P, V ) 1k = trace h Let {(ei , ei )}ni=1 ⊆ F ×F ∨ be dual bases for F = A⊕n reg , say the standard ones. Then we obtain dual bases {(xi , xi )}ni=1 for P by putting xi = π(ei ) and xi = ei ◦ µ. Chasing the idempotent h through the isomorphism ∼ Endk (HomA (F, V )) −→ Endk (V ⊕n )

∼ −→

Lemma 1.4

Matn (Endk (V ))


∼ coming from HomA (F, V ) −→ V ⊕n , f 7→ f (ei ) , one sees that h corresponds to the matrix (hi,j ) ∈ Matn (Endk (V )) that is given by hi,j (v) = hxj , xi iv for v ∈ V . Therefore,

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2 Further Topics on Algebras and their Representations

trace h =

X

trace hi,i =

i

= hχV ,

X

trace(hxi , xi iV )

i

X i

and the proof is complete.

i

hx , xi ii = hχV , rank P i (2.6)

t u

2.1.4 Finite-Dimensional Algebras In this section, we turn our attention to the case where the algebra A is finitedimensional. Then, of course, the categories A proj and Projfin A are identical and K0 (A) = P(A). Our first goal will be to study the indecomposable projectives of A. This will result in a more explicit description of the group P(A) than was offered in Proposition 2.4(a) and of the Cartan homomorphism c : P(A) → R(A) in (2.9). Lifting idempotents In this paragraph, we will discuss a purely ring theoretic lemma and the algebra A need not necessarily be finite-dimensional. First some terminology. An ideal I of A is called nil if all elements of I are nilpotent. A collection {ei }i∈I of idempotents of A is called orthogonal if ei ej = δi,j ei for i, j ∈ I. Lemma 2.6. Let I be a nil ideal of the algebra A and let {fi }n1 ⊆ A/I be a finite collection of orthogonal idempotents. Then there exist orthogonal idempotents ei ∈ A such that ei + I = fi . Proof. First, consider the case n = 1 and write f = f1 ∈ A/I. Let : A  A/I bdenote the canonical map and fix any a ∈ A such that a = f . Then the element b = 1 − a ∈ A satisfies ab = ba = a − a2 ∈ I, and hence (ab)m = 0 for some m ∈ N. Therefore, by the Binomial Theorem,
1 = (a + b)2m = e + e0 Pm 2m
2m−i i P2m 2m 2m−i i with e = b and e0 = b . By our choice of i=0 i=m+1 i a i a 0 0 0 2 m, we have ee = e e = 0 and so e = e(e + e ) = e is an idempotent. Finally, e ≡ a2m ≡ a mod I, whence e = a = f as desired. Note also that e is a polynomial in a with integer coefficients and zero constant term. Now let n > 1 and assume that we have already constructed e1 , . . . , en−1 ∈ A Pn−1 as in the lemma. Then x = i=1 ei is an idempotent of A such that ei x = xei = ei for 1 ≤ i ≤ n−1. Fix any a ∈ A such that a = fn and put a0 = (1−x)a(1−x) ∈ A. Pn−1 Then xa0 = a0 x = 0. Furthermore, since x = i=1 fi and a = fn are orthogonal idempotents of A/I, we have a0 = fn . Now construct the idempotent en ∈ A with en = fn from a0 as in the first paragraph. Since en is a polynomial in a0 with integer coefficients and zero constant term, it follows that xen = en x = 0. Therefore, ei en = ei xen = 0 and, similarly, en ei = 0 for i 6= n, completing the proof. t u

2.1 Projectives

87

Projective covers For any algebra A and any V ∈ Rep A, we have already repeatedly used the fact that there exists an epimorphism P  V with P projective or even free. For finitedimensional algebras (and some other algebras), there is a “minimal” choice for such an epimorphism which is in fact essentially unique. For any V ∈ Rep A, we consider the completely reducible factor head V = V /(rad A).V as in Exercise 1.32. This construction is functorial: any morphism φ : V → W in Rep A satifies φ((rad A).V ) ⊆ (rad A).W and hence φ induces a morphism head φ : head V → head W . Theorem 2.7. Let A be a finite-dimensional k-algebra. Then, for any V ∈ Rep A, there exists a P ∈ A Proj and an epimorphism φ : P  V satisfying the following equivalent conditions: (i) Ker φ ⊆ (rad A).P ; ∼ (ii) head φ : head P −→ head V ; (iii) Every epimorphism φ0 : P 0  V with P 0 ∈ some epimorphism π : P 0  P .

A Proj

factors as φ0 = φ ◦ π for

In particular, P is determined by V up to isomorphism. Proof. We start by proving the existence of an epimorphism φ satisfying (i). First assume that V is irreducible. Then V is a direct summand of the regular s.s representation of As.s = A/ rad A, and hence V ∼ = A f for some idempotent f ∈ s.s A . Since rad A is nil, even nilpotent, Lemma 2.6 guarantees the existence of an idempotent e ∈ A so that e = f under the canonical map : A  As.s . Putting P = Ae and φ = , we obtain a projective P ∈ A Proj and an epimorphism φ : P  V satisfying Ker φ = Ae ∩ rad A = (rad A)e = (rad A).P as required. L ⊕m(S,V ) Next assume that V is completely reducible and write V ∼ = S∈Irr A S as in (1.44). For each S, let φS : PS  S be the epimorphism constructed in the previous paragraph. Then the map M M ⊕m(S,V ) ∼ M ⊕m(S,V ) ⊕m(S,V ) : P =  φ= φS PS S −→ V S∈Irr A

S∈Irr A

S∈Irr A

satisfies the requirements of (i). For general V , consider the epimorphism φ : P  head V constructed in the previous paragraph. Then Proposition 2.1 yields a morphism φe as in the diagram P e ∃φ

V

can.

φ

head V

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2 Further Topics on Algebras and their Representations

Since φ = φe ◦ can is surjective, it follows that Im φe + (rad A).V = V . Iterating this equality, we obtain Im φe + (rad A)i .V = V for all i and hence Im φe = V , because (rad A)i = 0 for some i. Therefore, φe : P  V and, of course, Ker φe ⊆ Ker φ ⊆ (rad A).P . This completes the proof of the existence claim in the theorem. In order to prove the equivalence of (i) – (iii), note that φ : P  V induces an epimorphism head φ : head P  head V with Ker(head φ) = φ−1 ((rad A).V )/(rad A).P = (Ker φ + (rad A).P )/(rad A).P Therefore, head φ is an isomorphism if and only if Ker φ ⊆ (rad A).P , proving the equivalence of (i) and (ii). Now assume that φ satisfies (i) and let φ0 be as in (iii). Then Proposition 2.1 yields the diagram P0 ∃π

P

φ

φ

0

V

Exactly as above, it follows from surjectivity of φ0 that P = Im π + Ker φ. Thus, P = Im π + (rad A).P by (i) and iteration of this equality gives P = Im π. This shows that (i) implies (iii). For the converse, assume that φ satisfies (iii) and pick some epimorphism φ0 : P 0  V with P 0 ∈ A Proj and Ker φ0 ⊆ (rad A).P 0 . By (iii), there exists an epimorphism π : P 0  P with φ0 = φ ◦ π. Therefore, Ker φ = π(Ker φ0 ) ⊆ (rad A).π(P 0 ) = (rad A).P and so φ satisfies (i). This completes the proof of the equivalence of (i) – (iii). Finally, for uniqueness, let φ : Pπ  V and φ0 : P 0  V both satisfy (i) – (iii). Then there are epimorphisms P 0 P 0 such that φ ◦ π 0 = φ0 and φ0 ◦ π = φ. π Consequently, φ = φ ◦ π 0 ◦ π and so Ker π ⊆ Ker φ ⊆ (rad A).P . On the other hand, P = Q ⊕ Ker π for some Q, because the epimorphism π splits. Therefore, Ker π = (rad A). Ker π, which forces Ker π = 0 by nilpotency of rad A. Hence π is an isomorphism and the proof of the theorem is complete. t u The projective constructed in the theorem above for a given V ∈ Rep A is called the projective cover of V ; it will be denoted by P(V ) Thus, we have an epimorphism P(V )  V and P(V ) is minimal in the sense that P(V ) is isomorphic to a direct summand of every P ∈ A Proj such that P  V by (iii). Theorem 2.7(ii) states that head P(V ) ∼ = head V Exercise 2.12 explores some further basic properties of the operator P( · ).

(2.13)

2.1 Projectives

89

Principal indecomposable representations Let A be a finite-dimensional k-algebra. Then the regular representation Areg decomposes into a finite direct sum of indecomposable representations. The summands occurring in this decomposition are called the principal indecomposable representations of A; they evidently belong to Projfin A and they are unique up to isomorphism by the Krull-Schmidt Theorem (§1.2.6). It is easy to see that the principal indecomposable representations of A are exactly the indecomposable projectives of A (Exercise 2.13). The following proposition lists some further properties of the principal indecomposable representations. Recall that, for any finite-length V ∈ Rep A and any S ∈ Irr A, the multiplicity of S as a composition factor of V is denoted by µ(S, V ); see the Jordan-Hölder Theorem (Theorem 1.18). Proposition 2.8. Let A be a finite-dimensional k-algebra. Then: (a) The principal indecomposable representations of A are exactly the projective covers P(S) with S ∈ Irr A. Furthermore, M Areg ∼ P(S)⊕ dimD(S) S = S∈Irr A

(b) For any V ∈ Repfin A and any S ∈ Irr A, (P(S), V ) = µ(S, V ) dimk D(S) Proof. (a) Since head P(S) ∼ = S by (2.13), the various P(S) are pairwise non∼ isomorphic and they are all indecomposable. Furthermore, P(As.s reg ) = Areg , because s.s the canonical map A  A has L kernel rad A. Wedderburn’s Structure Theorem ⊕ dimD(S) S ∼ gives the decomposition As.s . The isomorphism in (a) reg = S∈Irr A S now follows by additivity of the operator P( · ) on direct sums (Exercise 2.12). The isomorphism also shows that the projective covers P(S) account for all principal indecomposable representations of A. This proves (a). For (b), note that the function (P(S), · ) = dimk HomA (P(S), · ) is additive on short exact sequences in Repfin A by exactness of the functor HomA (P(S), · ), and so is the multiplicity µ(S, · ) by (1.31). Therefore, by considering a composition series of V , one reduces to the case where V ∈ Irr A. But then µ(S, V ) = δS,V and HomA (P(S), V ) ∼ = HomA (head P(S), V ) ∼ = HomA (S, V ) = δS,V D(S) by Schur’s Lemma. The formula in (b) is immediate from this. t u As a special case of the multiplicity formula in Proposition 2.8(b), we note the so-called orthogonality relations: (P(S), S 0 ) = δS,S 0 dimk D(S)

(S, S 0 ∈ Irr A)

(2.14)

The multiplicity formula in Proposition 2.8(b) and the orthogonality relations (2.14) have a particularly simple and appealing form when the base field k is a splitting field for A, that is, D(S) = k for all S ∈ Irr A.

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2 Further Topics on Algebras and their Representations

The Cartan matrix We now return to the Grothendieck groups P(A) = K0 (A) and R(A). Both these groups are free abelian of finite rank equal to the size of Irr A. Indeed, the classes [S] ∈ R(A) with S ∈ Irr A provide a Z-basis of R(A) (Proposition 1.43) and the classes [P(S)] ∈ P(A) form a Z-basis of P(A) (Propositions 2.4(a) and 2.8(a)). In terms of these bases, the Cartan homomorphism c : P(A) → R(A) in (2.9) has the following description: c

∈

⊕ Irr A R(A) ∼ =Z

∈

⊕ Irr A P(A) ∼ =Z

P

[P(S)]

0

S ∈Irr A

µ(S 0 , P(S))[S 0 ]

Thus, the Cartan homomorphism can be described by the following integer matrix:   C = µ(S 0 , P(S)) 0 (2.15) S ,S∈Irr A

This matrix is called the Cartan matrix of A. Note that all entries of C belong to Z+ and that  the diagonalentries are strictly positive. If k is a splitting field for A, then C = (P(S 0 ), P(S)) 0 by Proposition 2.8(b). —> Exercises S ,S∈Irr A

Characters of projectives In this paragraph, we relate the character χP for P ∈ Projfin A to the HattoriStallings rank of P . In particular, we shall see that the latter determines the former. The reader is reminded that the character homomorphism χ : R(A) → A∗trace = (Tr A)∗ ∼ (Tr As.s )∗ of (Tr A)∗ ; see (1.55). has image in the subspace C(A) = Consider the regular (A, A)-bimodule A (Example 1.2). Let A a, bA ∈ Endk (A) denote right and left multiplication with a, b ∈ A, resp.: (bA ◦ A a)(x) = bxa. Define a k-linear map t = tA : A −→ A∗ by def

ht(a), bi = trace(bA ◦ A a)

(2.16)

Note that if a or b belong to [A, A], then bA ◦ A a ∈ [Endk (A), Endk (A)] and so trace(bA ◦ A a) = 0 . Moreover, if a or b belong to rad A, then the operator bA ◦ A a is nilpotent, and so trace(bA ◦ A a) = 0 again. Therefore, the map t factors through maps as in the following diagram; they will all be denoted by t: t

A

A∗

can.

Tr A

t

A∗trace = (Tr A)∗

t

(Tr As.s )∗ ∼ = C(A)

can.

Tr As.s

(2.17)

2.1 Projectives

91

The following proposition is due to Bass [9]. Proposition 2.9. Let A be a finite-dimensional k-algebra. Then the following diagram commutes: c

P(A)

R(A) χ

rank t

Tr A

(Tr A)∗

Proof. We need to show that, for P ∈ Projfin A and a ∈ A, hχP , ai = trace(aA ◦ A rank P ) To this end, fix dual bases {(bi , bi )}i ⊆ A × A∗ for A as k-vector space, and let {(xj , xj )}j ⊆ P × P ∨ be dual bases for P . Then {(bi .xj , bi ◦ xj )}i,j are dual bases for P over k: for p ∈ P , X j X i j X i p= hx , pi.xj = hb , hx , piibi .xj = hb ◦ xj , pibi .xj j

i,j

i,j

P Thus, IdP corresponds to i,j bi .xj ⊗ bi ◦ xj ∈ P ⊗ P ∗ under the standard isomor∗ ∼ phism P Endk (P ) =i P ⊗j P and, for any a ∈ A, the endomorphism aP corresponds to i,j abi .xj ⊗ b ◦ x . Therefore, hχP , ai = trace aP =

X i X i hb ◦ xj , abi .xj i = hb , abi hxj , xj ii i,j

i,j

X i = hb , abi rank P i = trace(aA ◦ A rank P )

(2.6)

i

t u

as claimed. The Hattori-Stallings rank map

Recall that, if k is a splitting field for A, then the character map yields an isomorphism of vector spaces (Proposition 1.46) s.s ∗ ∼ Rk (A) = R(A) ⊗Z k −→ C(A) ∼ = (Tr A )

Our goal in this paragraph is to prove a version of this result for P(A), with the Hattori-Stallings ranks replacing characters. This will further highlight the duality between P(A) and R(A). Let def

rankk : Pk (A) = P(A) ⊗Z k −→ Tr A denote the k-linear extension of the Hattori-Stallings rank map.

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2 Further Topics on Algebras and their Representations

Theorem 2.10. Let A be a finite dimensional k-algebra and let τ : Tr A  Tr As.s denote the canonical epimorphism. (a) If char k = 0, then τ ◦ rank is a group monomorphism P(A) ,→ Tr As.s . (b) If k is a splitting field for A, then we have a k-linear isomorphism ∼ τ ◦ rankk : Pk (A) −→ Tr As.s

Thus, the images of the principal indecomposable representations P(S) with S ∈ Irr A form a k-basis of Tr As.s . Proof. Put ρ = τ ◦ rankk : Pk (A) → Tr As.s . Then, for S, S 0 ∈ Irr A, we have hχS 0 , ρ[P(S)]i = hχS 0 , rank P(S)i

=

Proposition 2.5

(P(S), S 0 ) 1k

= δS,S 0 dimk D(S) 1k

(2.14)

Thus, the images ρ[P(S)] with dimk D(S) 1k 6= 0 form a k-linearly independent subset of Tr As.s . If char k = 0 or if k is a splitting field for A, then this holds for all S ∈ Irr A. Since Pk (A) is generated by the classes [P(S)] ⊗ 1 (Propositions 2.4(a) and 2.8(a)), we obtain a k-linear embedding ρ : Pk (A) ,→ Tr As.s in these cases. ⊕ Irr A ⊕ Irr A If char k = 0, then the canonical map P(A) ∼ → Pk (A) ∼ =Z =k s.s is an embedding, proving (a) For large k, we have dimk Tr A = dimk C(A) = # Irr A by Theorem 1.41 and (b) follows. t u

Exercises 2.1 (Projectives and semisimplicity). Let A be a k-algebra. Show that A is semisimple if and only if all V ∈ Rep A are projective. dn+1

d

n 2.2 (Exact functors). A sequence of morphisms · · · −→ Xn+1 −→ Xn −→ Xn−1 −→ · · · in A Mod is called a chain complex if dn ◦ dn+1 = 0 for all n or, equivalently, Im dn+1 ⊂ Ker dn . If equality holds here, for a given n, then the chain complex is said to be exact at Mn ; the chain complex is called exact if equality holds for all n. Functor A Mod → B Mod exact: preserves exactness of chain complexes; equivalent to: preserves exactness of short exact sequences fill this in.

2.3 (Injectives). Let A be an arbitrary k-algebra. A module I ∈ injective if I satisfies the following equivalent conditions:

A Mod

is called

(i) Given a monomorphism f : M ,→ N and an arbitrary g : M → I in A Mod, there exists a “lift” ge : N → I in A Mod such that ge ◦ f = g: I ∃g e

N

f

g

M

2.1 Projectives

93

(ii) Every monomorphism f : I ,→ M in A Mod splits: there exists s : M → I such that s ◦ f = IdI . (iii) The functor HomA ( · , I) : A Mod → Vectk is exact. (a) Prove the equivalence of the above conditions. (b) Let A → B be an algebra map. Show that coinduction CoindB A: Mod (§1.2.2) sends injectives of A to injectives of B. B

A Mod

→

2.4 (Some details for the Dual Bases Lemma). (a) Show that the image of the map (2.2) is an ideal of EndA (M ). (b) Work out a detailed proof of the injectivity claim in the Dual Bases Lemma (Lemma 2.2). 2.5 (Morita contexts). A Morita context consists of the following data: k-algebras A and B, bimodules V ∈ A ModB and W ∈ B ModA , and bimodule maps f : V ⊗B W → A and g : W ⊗A V → B, where A and B are viewed as the regular bimodules. The maps f and g are required to satisfy the following associativity conditions, for all v, v 0 ∈ V and w, w0 ∈ W : f (v⊗w).v 0 = v.g(w⊗v 0 ) and g(w⊗v).w0 = w.f (v⊗w0 ). Thus, using the multiplications of A and B, the actions of A and B on V and W , and the maps f and g to define a multiplication, the collection of generalized matrices A V ) becomes a k-algebra. Assuming g to be surjective, prove: (W B (a) g is an isomorphism; (b) Every left B-module is a homomorphic image of a direct sum of copies of W and every right B-module is a homomorphic image of a direct sum of copies of V ; (c) V and W are finitely generated projective as A-modules. 2.6 (Morita contexts and finiteness conditions). This problem assumes familiarity with Exercise 2.5 and uses the same notation. Let (A, B, V, W, f, g) be a Morita context such that A is right noetherian and g is surjective, prove: (a) B is right noetherian and V is finitely generated as right B-module. (b) If A is also affine, then B is affine as well. 2.7 (Some details for Hattori-Stallings traces). Verify the claim made in the proof of Lemma 2.3(b) and formula (2.8). 2.8 (Contravariant functoriality for projectives and Hattori-Stallings traces). Do this along the lines of Bass. 2.9 (Equality in K0 (A) and P(A) and stable isomorphism). (a) For P, Q ∈ A proj, show that [P ] = [Q] holds in K0 (A) if and only if P and Q are stably iso⊕r ∼ morphic: P ⊕ A⊕r reg = Q ⊕ Areg for some r ∈ Z+ . (b) Show that the homomorphism P(A) → K0 (A), [P ] 7→ [P ], is injective.
2.10 (Hattori-Stallings rank). Let A be an arbitrary algebra and let e = eij ∈ Matn (A) P be an idempotent matrix and let P = A⊕n reg · e as in (2.3). Show that rank P = i eii + [A, A].

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2 Further Topics on Algebras and their Representations

2.11 (Eilenberg swindle). (a) Let P be an arbitrary projective of some algebra A ⊕∞ and fix a free A-module F such that F = P 0 ⊕ Q with P 0 ∼ = = P . Show that F ⊕∞ ∼ ⊕∞ F ⊕ F ⊕ F ⊕ · · · is a free A-module satisfying P ⊕ F . =F (b) Let K0∞ (A) be constructed exactly as K0 (A) but using arbitrary projectives of A. Show that K0∞ (A) = {0}. 2.12 (Some properties of projective covers). Let A be a finite-dimensional algebra, let V, W ∈ Rep A, and let α : P(V )  V and β : P(W )  W . Prove: (a) If φ : V → W is a homomorphism in Rep A, then there exists a lift φ : P(V ) → P(W ) with φ ◦ α = β ◦ φ. Furthermore, φ is surjective if φ is surjective. (b) P(head V ) ∼ = P(V ). (c) P(V ⊕ W ) ∼ = P(V ) ⊕ P(W ). 2.13 (Indecomposable projectives). Let A be a finite-dimensional algebra. Show that every indecomposable projective P of A has the form P ∼ = P(S) for some S ∈ Irr A. 2.14 (Cartan matrix of the Sweedler algebra). Let char k 6= 2 and consider the algebra A = khx, yi/(x2 , y 2 − 1, xy + yx). This algebra is called the Sweedler algebra. (a) Realize A as a homomorphic image of the quantum plane Oq (k2 ) with q = −1 (Exercise 1.15) and use this to show that dimk A = 4. (b) Show that rad A = (x) and As.s ∼ = k × k. Specifically, there are two irreducible A-modules: k± with x 7→ 0, y 7→ ±1. (c) Show that e± = 21 (1 ± y) ∈ A are idempotents with A = Ae+ ⊕ Ae− and xe± = e∓ x. Conclude that P(k± ) = Ae∓ and that the Cartan matrix of A is C = ( 11 11 ).

2.2 Frobenius and Symmetric Algebras We conclude this chapter by discussing in some detail a special class of finitedimensional algebras, called Frobenius algebras, with particular emphasis on the subclass of symmetric algebras. As we will see in this section, all finite-dimensional semisimple algebras are symmetric, and it is in fact quite useful to think of semisimple algebras in this larger context. We will learn later that the class of Frobenius algebras encompasses all group algebras of finite groups and, more generally, all finite-dimensional Hopf algebras. The material developed in this section is admittedly somewhat technical; in the main, it consists of a collection of functions and formulas that will see some heavy use in Chapter 12. 2.2.1 Definition of Frobenius and Symmetric Algebras Recall from Example 1.2 that every k-algebra A carries the “regular” (A, A)bimodule structure: the left action of a ∈ A on A is given by the left multiplication

2.2 Frobenius and Symmetric Algebras

95

operator, aA , and the right action by right multiplication, A a. This structure gives rise to a bimodule structure on the linear dual A∗ = Homk (A, k) for which it is customary to use the following notation: a*f (b = f ◦ bA ◦ A a

(a, b ∈ A, f ∈ A∗ )

Using h · , · i : A∗ × A → k to denote the evaluation pairing, (A, A)-bimodule action becomes ha*f (b, ci = hf, bcai (a, b, c ∈ A, f ∈ A∗ ) (2.18) The algebra A is said to be Frobenius if A∗ , viewed as a left A-module, is isomorphic to the left regular A-module A A. We will see in Lemma 2.13 below that this condition is equivalent to corresponding right A-module condition. If A∗ and A are in fact isomorphic as (A, A)-bimodules – this is not automatic from the existence of a one-sided module isomorphism – then the algebra A is called symmetric. Note that even a mere k-linear isomorphism A∗ ∼ = A forces A to be finite-dimensional (Appendix B); so Frobenius algebras will necessarily have to be finite-dimensional. Here are some immediate consequence of the definition of Frobenius algebras. Remark 2.11 (The regular character of a Frobenius algebra). Due to the switch in sides when passing from A to A∗ in (2.18), the left and right regular representation of any Frobenius algebra A have the same character. Indeed, for any a ∈ A, we compute trace(aA ) = trace((aA )∗ ) = trace(A∗ a) = trace(A a) (B.24)

where the last equality uses the fact that A∗ ∼ = A as right A-modules. Later in this section, we will derive a more specific expression for the regular character χreg that will also imply the above equality. Remark 2.12 (Self-injectivity). Note that the above left A-module structure on A∗ ∼ is identical to the one that was denoted by CoindA k k in §1.2.2. Thus, Areg = A Coindk k. Since coinduction preserves injectivity (Exercise 2.3), we obtain that the regular representation Areg is injective. — Consolidate with Exercise 2.16: A Proj ≡ A Inj when A is Frobenius. 2.2.2 Frobenius Form, Dual Bases and Nakayama Automorphism For a finite-dimensional algebra A, the existence of a left A-module isomorphism ∗ ∗ A∗ ∼ = A amounts to the existence of an element λ ∈ A such that A = A*λ; similarly for the right A-module structures. The next lemma shows in particular that any left A-module generator λ ∈ A∗ also generates A∗ as right A-module and conversely.

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2 Further Topics on Algebras and their Representations

Lemma 2.13. Let A be a finite-dimensional k-algebra. Then, for any linear form λ ∈ A∗ , the following are equivalent: (i) A∗ = A*λ (ii) There exist elements xi , yi ∈ A (i = 1, . . . , dimk A) such that X a= xi hλ, ayi i for all a ∈ A

(2.19)

i

or, equivalently, a=

X

yi hλ, xi ai

for all a ∈ A

(2.20)

i

(iii) A∗ = λ(A Proof. Observe that condition (2.19) states that {xi } is a k-basis of A and {yi *λ} is the corresponding dual basis of A∗ . This certainly implies that λ generates A∗ as left A-module; so (i) holds. Conversely, if (i) is satisfied, then for any k-basis {xi } of A, the dual basis of A∗ has the form {yi *λ} for suitable yi ∈ A, giving (2.19). Similarly, (2.20) states that {(yi , λ(xi )} are dual bases of A and A∗ , and the existence of such bases is equivalent to (iii). Finally, since hyi *λ, xj i = hλ, xj yi i = hλ(xj , yi i, we see that {(yi , λ(xi )} are dual bases if and only if {(xi , yi *λ)} are. Hence, the two conditions (2.19) and (2.20) are equivalent to each other, proving the lemma. t u Thus, a finite-dimensional algebra A is a Frobenius algebra if and only if there is a linear form λ ∈ A∗ satisfying the equivalent conditions of Lemma 2.13; any such λ is called a Frobenius form. Note that the equality A∗ = A*λ in (a) is equivalent to the condition that 0 6= a ∈ A implies λ(Aa) 6= 0, which in turn is equivalent to the corresponding condition for aA by Lemma 2.13. Thus, a Frobenius form is a linear form λ ∈ A∗ such that Ker λ contains no nonzero left ideal or, equivalently, no nonzero right ideal of A. We will think of a Frobenius algebra as the pair (A, λ) A homomorphism of Frobenius algebras f : (A, λ) → (B, µ) is an algebra map such that µ ◦ f = λ. The elements (xi , yi ) in Lemma 2.13(b) are called dual bases of A with respect to λ. The identities (2.19) and (2.20) can be expressed by the diagram ∼

A ⊗ A∗ ∈

can.

∈

Endk (A) IdA

xi ⊗ (yi *λ) = yi ⊗ (λ(xi )

(2.21)

Here, we have dispensed with the summation symbol, and we shall continue to do so below: Summation over indices occurring twice is implied throughout this section.

2.2 Frobenius and Symmetric Algebras

97

For a given Frobenius form λ ∈ A∗ and a ∈ A, Lemma 2.13 implies that λ(a = νλ (a)*λ

(2.22)

for a unique νλ (a) ∈ A. Thus, hλ, abi = hλ, bνλ (a)i

(a, b ∈ A)

This determines an automorphism νλ ∈ AutAlgk (A), the Nakayama automorphism of A that is associated to λ: νλ (a) = yi hλ, xi νλ (a)i = yi hλ, axi i (2.20)

(2.23)

As an application, we mention another left-right symmetry property of the regular representation of any Frobenius algebra. Example 2.14 (The socle series of a Frobenius algebra). The left and right regular representation of any Frobenius algebra A have the same socle: l. annA (rad A) = r. annA (rad A). In fact, more generally, we have def

socn A = l. annA (rad A)n = r. annA (rad A)n This follows from the observation that, for any ideal I of A and any a ∈ A, the defining property of a Frobenius form implies that Ia = 0 ⇔ hλ, Iai = 0 ⇐⇒ hλ, aνλ (I)i = 0 ⇔ aνλ (I) = 0 (2.22)

So r. annA I = l. annA νλ (I). Since rad A and all its powers are stable under νλ , or any other automorphism of A, we obtain the asserted equality for socn A. Changing the Frobenius form The data associated to A that we have assembled above are unique up to units. Indeed, for each unit u ∈ A× , the form u*λ ∈ A∗ is also a Frobenius form and all Frobenius forms of A arise in this way, because they are just the possible generators of the left A-module A A∗ ∼ = A A. The Nakayama automorphisms that are associated to λ and to λ0 = u*λ are related by νλ0 (a) = uνλ (a)u−1

(a ∈ A)

(2.24)

as the reader will easily ascertain. If (xi , yi ) are dual bases of A with respect to λ — so hλ, xi yj i = δi,j — then (xi , yi u−1 ) are dual bases with respect to u*λ.

98

2 Further Topics on Algebras and their Representations

2.2.3 Casimir Elements, Casimir Operator and Higman Trace Let (A, λ) be a Frobenius algebra. The elements of A ⊗ A that correspond to IdA ∗ under the two isomorphism Endk (A) ∼ = A⊗A ∼ = A ⊗ A that are obtained by ∗ identifying A and A via · *λ and λ( · will be referred to as the Casimir elements * ( that are associated to the Frobenius form λ; they will be denoted by cλ and cλ , respectively. In later chapters, we will consider similar elements, also called Casimir elements, for semisimple Lie algebras ((5.54) and §6.2.1). By (2.21), the Casimir elements are given by *

cλ = xi ⊗ yi (

(

cλ = yi ⊗ xi

and

(2.25)

*

Thus cλ = τ (cλ ), where τ ∈ AutAlgk (A ⊗ A) is the switch map, τ (a ⊗ b) = b ⊗ a. The Casimir elements do depend on λ but not on the choice of dual bases (xi , yi ). Lemma 2.15. Let (A, λ) be a Frobenius algebra with Nakayama automorphism νλ ∈ AutAlgk (A). Then the following identities hold in the algebra A ⊗ A: *

(

*

(

*

(

(a) cλ = (Id ⊗νλ )(cλ ) = (νλ ⊗ νλ )(cλ ) and cλ = (νλ ⊗ Id)(cλ ) = (νλ ⊗ νλ )(cλ ). * * ( ( (b) (a ⊗ b)cλ = cλ (b ⊗ νλ (a)) and (a ⊗ b)cλ = cλ (νλ (b) ⊗ a) for all a, b ∈ A. Proof. The second identities in (a) and (b) follow from the first by applying τ ; so we * will focus on the formulas for cλ . * ( (a) The equality cλ = (Id ⊗νλ )(cλ ) follows from
xi ⊗ (yi *λ) = yi ⊗ (λ(xi ) = yi ⊗ νλ (xi )*λ (2.21)

(2.22)

*

(

Applying νλ ⊗ Id, we obtain (νλ ⊗ Id)(cλ ) = (νλ ⊗ νλ )(cλ ) and then τ yields ( * (Id ⊗νλ )(cλ ) = (νλ ⊗ νλ )(cλ ). (b) We compute axi ⊗ yi = xj hλ, axi yj i ⊗ yi = xj ⊗ yi hλ, xi yj νλ (a)i = xj ⊗ yj νλ (a) (2.19)

(2.22)

(2.20)

xi ⊗ byi = xi ⊗ yj hλ, xj byi i = xi hλ, xj byi i ⊗ yj = xj b ⊗ yj (2.20)

(2.19)

t u

This gives the first equality in (b).

Continuing with the notation of Lemma 2.15, we will now discuss two closely related operators that were originally introduced by Donald Higman [83]: (

a

xi ayi

and

ZA

A ∈

γλ : A

A ∈

Tr(A)

∈

Tr

∈

*

γλ : A

a

yi axi

(2.26)

2.2 Frobenius and Symmetric Algebras *

99

(

The operator γλ will be called the Higman trace and γλ will be referred to as the Casimir operator. The following lemma justifies the claims, implicit in (2.26), that the Higman trace does indeed factor through the universal trace Tr : A  Tr(A) = A/[A, A] and the Casimir operator takes values in the center Z A. If some of this is not actually used, make exercises. Lemma 2.16. Let (A, λ) be a Frobenius algebra with Nakayama automorphism νλ ∈ AutAlgk (A). Then: *

*

(

(

(a) νλ ◦ γλ = γλ ◦ νλ and νλ ◦ γλ = γλ ◦ νλ . * * ( ( (b) For all a, b, c ∈ A, we have aγλ (bc) = γλ (cb)νλ (a) and aγλ (bc) = γλ (νλ (c)b)a. *2 * * (2 * ( (c) cλ = (Id ⊗γλ )(cλ ) and cλ = (γλ ⊗ Id)(cλ ) Proof. Once again, we shall only prove the first identities in (a) – (c), since the proof of the second is analogous or follows by applying τ . * * (a) The identity cλ = (νλ ⊗ νλ )(cλ ) in Lemma 2.15(a) gives
* * γλ (a) = xi ayi = νλ (xi )aνλ (yi ) = νλ (xi νλ−1 (a)yi ) = νλ ◦ γλ ◦ νλ−1 (a) *

*

This shows that νλ ◦ γλ = γλ ◦ νλ . (b) The first identity in Lemma 2.15(b) states that axi ⊗ byi = xi b ⊗ yi νλ (a). Multiplying this on the right with c ⊗ 1 and then applying the multiplication map * * A ⊗ A → A gives axi cbyi = xi bcyi νλ (a) or, equivalently, aγλ (cb) = γλ (bc)νλ (a). * * (c) From the identity cλ (xi ⊗ 1) = (1 ⊗ xi )cλ in Lemma 2.15(b) we obtain *2

*

*

*

*

cλ = cλ (xi ⊗ yi ) = (1 ⊗ xi )cλ (1 ⊗ yi ) = (Id ⊗γλ )(cλ ) t u 2.2.4 Trace Formulas We now apply the foregoing to derive some explicit trace formulas that will be useful later on. Lemma 2.17. Let (A, λ) be a Frobenius algebra with dual bases (xi , yi ). Then, for any f ∈ Endk (A), trace(f ) = hλ, f (xi )yi i = hλ, xi f (yi )i Proof. By (2.21), we have ∼

A ⊗ A∗ ∈

can.

∈

Endk (A) f

f (xi ) ⊗ (yi *λ) = f (yi ) ⊗ (λ(xi )

Since the trace function on Endk (A) becomes evaluation on A ⊗ A∗ , we obtain the formula in the lemma. t u

100

2 Further Topics on Algebras and their Representations

With f = bA ◦ A a for a, b ∈ A, Lemma 2.17 yields
the following expression for the map tA : A → A∗ , a 7−→ b 7→ trace(bA ◦ A a) from (2.16) in terms of the Higman trace: * * trace(bA ◦ A a) = hλ, bγλ (a)i = hλ, γλ (b)ai (2.27) Equation (2.27) shows in particular that the left and right regular representation of A have the same character, as we have already observed earlier (Remark 2.11): χreg (a) = trace(aA ) = trace(A a) *

*

(2.28)

*

= hλ, γλ (a)i = hλ, γλ (1)ai = hλ, aγλ (1)i *

(

Proposition 2.18. Let (A, λ) be a Frobenius algebra. Then γλ and γλ both vanish on rad A and their images are contained in soc A . Proof. As we have already observed in (2.17), the operator bA ◦ A a ∈ Endk (A) is nilpotent if a or b belong to rad A, and hence trace(bA ◦ A a) = 0. Consequently, * * (2.27) gives hλ, Aγλ (rad A)i = 0 and hλ, rad A · γλ (A)i = 0. Since the Frobenius * form λ does not vanish on nonzero left ideals, we must have γλ (rad A) = 0 and * * * rad A · γλ (A) = 0. This shows that rad A ⊆ Ker γλ and Im γλ ⊆ soc A. ( For γλ , we first compute (

hλ, bγλ (a)i = hλ, byi axi i = hλ, xi νλ (byi a)i (2.22)

=

Lemma 2.17

trace(νλ ◦ bA ◦ A a)

The operator νλ ◦ bA ◦ A a ∈ Endk (A) is again nilpotent if a or b belong to rad A, because its nth power has image in (rad A)n in this case. We can now repeat the ( above reasoning verbatim to conclude that γλ also vanishes on rad A and has image in soc A. t u 2.2.5 Symmetric Algebras Recall that the algebra A is symmetric if there is an (A, A)-bimodule isomorphism ∗ ∗ ∼ A AA −→ A A A . In this case, the image of 1 ∈ A will be a Frobenius form λ ∈ A such that a*λ = λ(a holds for all a ∈ A; in other words, the associated Nakayama automorphism νλ is IdA or, equivalently, λ is a trace form. Recall also that Frobenius forms λ ∈ A∗ are characterized by the condition that Ker λ contains no nonzero left or right ideal of A. For λ ∈ Tr(A)∗ , this amounts to saying that Ker λ contains no nonzero two-sided ideal of A, because λ(Aa) = λ(AaA) = λ(aA) for all a ∈ A. In sum, a finite-dimensional algebra A is symmetric if and only if there is a trace form λ ∈ Tr(A)∗ such that Ker λ contains no nonzero ideal of A. In light of (2.24), a symmetric algebra is the same as a Frobenius algebra A possessing a Frobenius form λ ∈ A∗ such that νλ is an inner automorphism of A, in which case the same holds for any Frobenius form of A. However, it will be convenient to always choose our Frobenius form λ ∈ Tr(A)∗ – this determines λ up to a central unit of A; see §2.2.2.

2.2 Frobenius and Symmetric Algebras

101

Casimir trace and regular character Let us note some consequences of the choice of a trace form λ ∈ Tr(A)∗ as Frobenius form. First, as we have already observed, νλ = IdA *

(

Thus, by Lemma 2.15(a), the two Casimir elements coincide, cλ = cλ , and the * ( Casimir operator is the same as the Higman trace, γλ = γλ . We will simply write cλ and γλ , respectively, and refer to γλ as the Casimir trace associated to λ. Thus, if (xi , yi ) are dual bases for λ, then the Casimir element and trace are given by cλ = xi ⊗ yi = yi ⊗ xi

and

γλ (a) = xi ayi = yi axi

(2.29)

First examples of symmetric algebras We start with a general observation. Proposition 2.19. Every finite-dimensional semisimple k-algebra is symmetric. Proof. Let A be semisimple. Wedderburn’s Structure Theorem allows us to assume that A is in fact simple, because a finite direct product of algebras is symmetric if all its components are (Exercise 2.18). By Theorem 1.41(b), we also know that Tr(A)∗ 6= 0. Fix any nonzero trace form λ. Then Ker λ contains no nonzero ideal of A, by simplicity, and so λ is a Frobenius form for A. t u Next, we offer some prototypes of symmetric algebras along with their Casimir traces and regular characters. The first example spells out Proposition 2.19 more explicitly for the special case of matrix algebras; the second example finally brings us back to finite group algebras, showing that they do indeed fit into the context of this section. Example 2.20 (Matrix algebras). If A = Matn (k) is the n × n-matrix algebra, then we can take the ordinary trace λ = trace ∈ Tr(A)∗ as Frobenius form. Dual bases for this form are provided by the standard matrices el,m , with 1 in the (l, m)-position and 0s elsewhere: trace(ej,k ek0 ,j 0 ) = δ(j,k),(j 0 ,k0 ) . Thus, the Casimir element is ctrace = ej,k ⊗ ek,j where and summation over
both j and k is implied. By (2.29), the Casimir trace of a matrix a = al,m ∈ A is γλ (a) = ej,k aek,j = ak,k ej,j ; so γtrace (a) = trace(a)1n×n . Identifying the center of A with k, we may write this as γtrace = trace Now (2.28) gives the following formula for the regular character, which was already observed much earlier (Exercise 1.45): χreg (a) = n trace(a)

(a ∈ A)

102

2 Further Topics on Algebras and their Representations

2.2.6 Semisimple Algebras as Symmetric Algebras Now assume that A is a split semisimple k-algebra and consider the Wedderburn isomorphism (1.46): A

∼

Endk (S) ∼ =

Y

Matdimk S (k)

∈

S∈Irr A

∈

S∈Irr A

Y

a

aS




Recall that the primitive central idempotent e(S) ∈ Z A is the element correspondQ ing to (0, . . . , 0, Id , 0, . . . , 0) ∈ S S∈Irr A Endk (S) under this isomorphism; so L Z A = S∈Irr A ke(S) and e(S)T = δS,T IdS

(S, T ∈ Irr A)

Our goal here is to give a formula for e(S) using Frobenius data of A – recall that A is symmetric (Proposition 2.19). We will also describe the image of the Casimir square c2λ ∈ A ⊗ A under the following isomorphism, coming from the Wedderburn isomorphism, A⊗A ∼

Y

Endk (S) ⊗ Endk (T )

∈

∈

S,T ∈Irr A

a⊗b

(a ⊗ b)S,T := aS ⊗ bT

(2.30)


For each c ∈ Z A and S ∈ Irr A, the operator cS ∈ Endk (S) is a scalar. Note that c2λ ∈ Z A ⊗ Z A, because (a ⊗ b)cλ = cλ (b ⊗ a) for all a, b ∈ A by Lemma 2.15(b). We shall be particularly interested in the value γλ (1)S ∈ k, where γλ is the Casimir trace (2.29), and in the matrix (c2λ )S,T ∈ MatIrr A (k). Theorem 2.21. Let A be a split semisimple k-algebra with Frobenius form λ ∈ Tr(A)∗ . (a) For each S ∈ Irr A , we have the following formula in A = k ⊗ A: e(S) γλ (1)S = (dimk S) (χS ⊗ IdA )(cλ ) = (dimk S) (IdA ⊗χS )(cλ ) In particular, γλ (1)S = 0 if and only if (dimk S) 1k = 0 . (b) (cλ )S,T = 0 for S 6= T ∈ Irr A and (dimk S)2 (c2λ )S,S = γλ (1)2S Proof. (a) The equality (χS ⊗ IdA )(cλ ) = (IdA ⊗χS )(cλ ) follows from (2.29). In order to show that e(S) γλ (1)S = (dimk S) (IdA ⊗χS )(cλ ), we use (2.19) to write e(S) = xi hλ, e(S)yi i, and we need to show that hλ, e(S)yi i γλ (1)S = (dimk S) χS (yi ) for all i or, equivalently,

2.2 Frobenius and Symmetric Algebras

hλ, ae(S)iγλ (1)S = (dimk S) χS (a)

103

(2.31)

for all a ∈ A. For this, we use the regular character: χreg (ae(S)) = hλ, ae(S)γλ (1)i = hλ, ae(S)γλ (1)S i = hλ, ae(S)iγλ (1)S (2.28)

⊕ dimk T ∼L from Wedderburn’s On the other hand, the isomorphism P Areg = T ∈Irr A T Structure Theorem gives χreg = T ∈Irr A (dimk T )χT . Since e(S)*χT = δS,T χS , we obtain e(S)*χreg = (dimk S)χS (2.32)

Therefore, χreg (ae(S)) = (dimk S)χS (a), proving (2.31). Since χS and hλ, e(S) · i are nonzero linear forms on A, equation (2.31) also shows that γλ (1)S = 0 if and only if (dimk S) 1k = 0. (b) For S 6= T , the identity (a ⊗ b)cλ = cλ (b ⊗ a) from Lemma 2.15(b) gives

(cλ )S,T = (e(S)⊗e(T ))cλ S,T = cλ (e(T )⊗e(S)) S,T = (cλ )S,T (0S ⊗0T ) = 0 It remains to consider the case S = T . For c ∈ Z (A), the operator cS ∈ Endk (S) is a scalar and χS (c) = (dimk S)cS . Therefore, writing ρS : A → Endk (S) for the representation given by S, we calculate, for any a ∈ A, (dimk S)(ρS ◦ γλ )(a) = (χS ◦ γλ )(a) = χS (xi ayi ) = χS (ayi xi ) = χS (a γλ (1)) = χS (a) γλ (1)S Using the formula c2λ = (γλ ⊗ Id)(cλ ) from Lemma 2.16(c), we further obtain
(dimk S)2 (c2λ )S,S = (dimk S)2 (ρS ⊗ ρS ) (γλ ⊗ Id)(cλ )
= (dimk S)2 (ρS ◦ γλ ) ⊗ ρS (cλ ) = (dimk S) (χS ⊗ ρS )(cλ ) γλ (1)S
= (Idk ⊗ρS ) (dimk S) (χS ⊗ Id)(cλ ) γλ (1)S
= ρS e(S)γλ (1)S γλ (1)S = γλ (1)2S (a)

which completes the proof of the theorem.

t u

2.2.7 Integrality and Divisibility Theorem 2.21 is a useful tool in proving certain divisibility results for the degrees of irreducible representations. For this, we recall some standard facts about integrality; proofs can be found in most textbooks on commutative algebra or algebraic number theory. Let R be a ring and let S be a subring of the center Z R. An element r ∈ R is said to be integral over S if r satisfies some monic polynomial over S. The following basic facts will be referred to repeatedly in later sections as well:

104

• • •

2 Further Topics on Algebras and their Representations

An element r ∈ R is integral over S if and only if r ∈ R0 for some subring R0 ⊆ R such that R0 contains S and is finitely generated as an S-module. If R is commutative, then the elements of R that are integral over S form a subring of R containing S; it is called the integral cosure of S in R. The integral closure of Z in Q is Z; in other words, an element of Q that is integral over Z must belong to Z.

The last fact above reduces the problem of showing that a given nonzero integer s divides another integer t to proving that the fraction st is merely integral over Z. Corollary 2.22. Let A be a split semisimple algebra over a field k of characteristic 0 and let λ ∈ Tr(A)∗ be a Frobenius form such that γλ (1) ∈ Z . Then the following are equivalent: (i) The degree of every irreducible representation of A divides γλ (1); (ii) the Casimir element cλ is integral over Z. Proof. Theorem 2.21 gives the formula (c2λ )S,S =



γλ (1) dimk S

2

(2.33)

Q If (i) holds, then the isomorphism (2.30) sends Z[c2λ ] to S∈Irr H Z, because (cλ )S,T vanishes for S 6= T by Theorem 2.21. Thus, Z[cλ ] is a finitely generated Z-module and (ii) follows. Conversely, (ii) implies that c2λ also satisfies a monic polynomial γλ (1) over Z and all (c2λ )S,S satisfy the same polynomial. Therefore, the fractions dim kS must be integers, proving (i). t u Corollary 2.23. Let A be a split semisimple algebra over a field k of characteristic 0 and let λ ∈ Tr(A)∗ be a Frobenius form for A. Furthermore, let (B, µ) be a Frobe* nius k-algebra such that γµ (1) ∈ k and let φ : (A, λ) → (B, µ) be a homomorphism with λ = µ ◦ φ. Then, for all S ∈ Irr A, *

γµ (1) B

dimk IndA S

=

γλ (1)S dimk S

If the Casimir element cλ is integral over Z, then so is the scalar

* γµ (1) B

dimk IndA S

∈ k.

B ⊕ dimk S ∼ Proof. Putting e := e(S), we have S ⊕ dimk S ∼ = = Ae and so IndA S Bφ(e). Since φ(e) ∈ B is an idempotent, we have dimk Bφ(e) = trace(B φ(e)) (Exercise 1.44). Therefore, *

*

⊕ dimk S dimk IndB = trace(B φ(e)) = hµ, φ(e)γµ (1)i = hµ, φ(e)iγµ (1) AS (2.28)

2

*

= hλ, eiγµ (1) =

(2.31)

The claimed equality rem 2.21 gives

* γµ (1) B

dimk IndA S *
2 γµ (1) B

dimk IndA S

=

γλ (1)S dimk S

(dimk S) * γ (1) γλ (1)S µ

is immediate from this. Finally, Theo-

= (c2λ )S,S , which is integral over Z if cλ is.

t u

2.2 Frobenius and Symmetric Algebras

105

2.2.8 Separability A finite-dimensional k-algebra A is called separable if K ⊗ A is semisimple for every field extension K/k. The reader is referred to Exercises 1.42 and 1.49 for more on separable algebras. Here, we give a characterization of separability in terms of the Casimir operator. For any Frobenius algebra A with Frobenius for λ ∈ A∗ , the ( ( Casimir operator γλ : A → Z A is Z A-linear. Hence, the image γλ (A) is an ideal of Z (A). This ideal does not depend on the choice of Frobenius form λ; indeed, if ( ( λ0 ∈ A∗ is another Frobenius form, then γλ0 (a) = γλ (ua) for some unit u ∈ A× (§2.2.2). Thus, we may define def (

Γ (A) = γλ (A) where λ ∈ A∗ is any Frobenius form for A. This ideal will be called the Casimir ideal of Z A. The following theorem is due to Donald Higman [83]. Theorem 2.24. The following are equivalent for a (finite-dimensional) k-algebra A. (i) A is separable; (ii) A is symmetric and Γ (A) = Z A; (iii) A is Frobenius and Γ (A) = Z A. Proof. The proof of (i) ⇒ (ii) elaborates on the proof of Proposition 2.19; we need to make sure that the current stronger separability hypothesis on A also gives Γ (A) = Z A. As in the earlier proof, Exercise 2.18 allows us to assume that A is simple. Thus, F := Z A is a field and F/k is a finite separable field extension (Exercise 1.49). It suffices to show that Γ (A) 6= 0. For this, let F denote an algebraic closure of F . Then A ⊗F F ∼ = Matn (F ) for some n (Exercise 1.13). The ordinary trace map trace : Matn (F ) → F is nonzero on A, since A generates Matm (F ) as F -vector space. It is less clear, that the restriction of the trace map to A has values in F , but this is in fact the case, giving a trace form tr : A  F This map is called the reduced trace of the central simple F -algebra A; see Reiner [152, (9.3)]. Since F/k is finite separable field extension, we also have the field trace TF/k : F  k — this is the same as the regular character χreg of the k-algebra F (Exercise 1.48). The composite λ := TF/k ◦ tr : A  k gives a nonzero trace form for the k-algebra A, which we may take as our Frobenius form. If (ai , bi ) are dual F -bases of A for tr and (ej , fj ) are dual k-bases of F for TF/k , then (ej ai , bi fj ) are easily checked to be dual k-bases of A for λ: hλ, ej ai bi0 fj 0 i = δ(i,i0 ),(j,j 0 ) . Moreover, γtr = tr by Example 2.20, and so Γ (A) = ej ai A bi fj = (γTF /k ◦ γtr )(A) = (γTF /k ◦ tr)(A) = γTF /k (F ) This is nonzero, because TF/k ◦ γTF /k = χreg = TF/k by (2.28).

106

2 Further Topics on Algebras and their Representations

The implication (ii) ⇒ (iii) being trivial, let us turn to the proof of (iii) ⇒ (i). Here, we can be completely self-contained. Note that the properties in (iii) are preserved under any field extension K/k: If λ ∈ A∗ is a Frobenius form for A such that ( γλ (a) = 1 for some a ∈ A, then λK = IdK ⊗λ is a Frobenius form for K ⊗ A – any ( ( pair of dual bases (xi , yi ) for λ also works for λK – and γλK (a) = γλ (a) = 1. Thus, ( it suffices to show that (iii) implies that A is semisimple. But 1 = γλ (a) ∈ soc A by Proposition 2.18. Hence soc A = A, proving that A is semisimple. t u

Exercises 2.15 (Center and twisted trace forms). Let (A, λ) be a Frobenius algebra. Show ∼ ∼ that the isomorphism λ( . : A −→ A∗ restricts to a (k-linear) isomorphism Z A −→ ∗ {f ∈ A | f (x = νλ (x)*f for all x ∈ A}. In particular, if (A, λ) is symmetric ∼ with λ ∈ Tr(A)∗ , then we obtain an isomorphism Z A −→ Tr(A)∗ . 2.16 (Frobenius algebras and Frobenius extensions). Show that A is a Frobenius k-algebra if and only if the unit map u : k → A is a Frobenius extension as in Exercise 2.22. Show that A Proj ≡ A Inj when A is Frobenius. — Consolidate with Exercise 2.22 *2

*

(

2.17 (Casimir operator and Higman trace). Show that γλ (a) = γλ (a)γλ (1) and (2 ( ( γλ (a) = γλ (a)γλ (1). *

*2

[Hint: Both operators are Z (A)-linear. Also, the trace property of γλ gives γλ (a) = * * * ( γλ (xi ayi ) = γλ (ayi xi ) = γλ (a)γλ (1). ]

2.18 (Stability). Prove: (a) The direct product A1 × A2 is Frobenius if and only if both A1 and A2 are Frobenius; similarly for symmetric. Furthermore, Γ (A1 × A2 ) = Γ (A1 ) × Γ (A2 ). (b) If A is Frobenius, then so is Matn (A); similarly for symmetric. Furthermore, Γ (Matn (A)) = Matn (Γ (A)).
∗ [Hint for (b): Any λ ∈ A yields a linear form for Matn (A) by defining λn (ai,j ) := i λ(ai,i ). Show that if λ is a Frobenius form, then so is λn . ]

P

2.19 (Separable algebras are symmetric). Separable algebras are symmetric; Reiner Theorem (9.26) — done. Explain CR1 p. 196; “Killing form” for associative algebras ?? 2.20 (Traces). Let A be a Frobenius algebra with Frobenius form λ and dual bases {(xi , yi )}. Show: (a) trace(aA ◦ A b) = trace(bA ◦ A a) for all a, b ∈ A. ????? — OK if A is symmetric: choose λ ∈ Tr(A)∗ in (2.27). (b) ??

2.2 Frobenius and Symmetric Algebras

107

2.21 (Reynolds operators). Let B be an arbitrary k-algebra and let A be a subalgebra of B. A Reynolds operator for the extension A ⊆ B, by definition, is an (A, A)-bimodule map π : B → A such that π|A = IdA . (The map πH that was employed in the proof of Proposition 3.3 is a Reynolds operator for kH ⊆ kG.) Reynolds operators are also referred to as conditional expectations. Assuming that such a map π exists, prove: (a) If B is left (right) noetherian, then so is A; similarly for “artinian”. B A (b) For every W in Rep A, the map IndB A π = π ⊗A IdW : IndA W → IndA W , A ∼ followed by the canonical isomorphism IndA W −→ W , yields a morphism of Arepresentations B φπ : ResB A IndA W → W B that splits the canonical map σ : W → ResB A IndA W , w 7→ 1 ⊗ w, in the sense that φπ ◦ σ = IdW . Conclude that W is isomorphic to a direct summand of B ResB A IndA W . ∗ A B (c) Similarly, the map CoindB A π = π : CoindA W → CoindA W , preceded by A ∼ the canonical isomorphism W −→ CoindA W , gives a map of A-representations, B ψπ : W → ResB A CoindA W B that splits the canonical map τ : ResB A CoindA W → W , f 7→ f (1), in the sense that τ ◦ ψπ = IdW . Furthermore, the unique lift of ψπ to a map of B representations B Ψπ : IndB A W → CoindA W (Proposition 1.9) satsfies τ ◦ Ψπ ◦ σ = IdW .

2.22 (Frobenius extensions). Let φ : A → B be a k-algebra map, also referred to as an algebra extension B/A, and view B as (A, A)-bimodule via φ as in §1.2.2: a.b.a0 = φ(a)bφ(a0 ). The extension B/A is called a Frobenius extension if there exists an (A, A)-bimodule map E : B → A and elements {xi }ni=1 , {yi }ni=1 of B such that, for all b ∈ B, X X b= xi .E(yi b) = E(bxi ).yi i

i

(The map πH in the proof of Proposition 3.3 works for kH ,→ kG.) (a) Mimic the proof of Proposition 3.3 to show that, for any Frobenius extension B ∼ B/A, there is an isomorphims of functors IndB A = CoindA . B ∼ (b) Conversely, if IndB A = CoindA , then B/A is a Frobenius extension. 2.23 (The Cartan matrix of a symmetric algebra). Let A be a symmetric k-algebra and assume that k is large enough for A. Show that the Cartan matrix of A is symmetric.

Part II

Groups

3 Groups and Group Algebras

The theory of group representations is the archetypal representation theory. A representation of a group G, by definition, is a group homomorphism G → GL(V ) where V is a k-vector space over a field k and GL(V ) = Autk (V ) denotes the group of all k-linear automorphisms of V . More precisely, such a representation is called a linear representation of G over the field k. Since the base field k is not part of the defining data of G, it can be chosen depending on the purpose at hand. Traditionally, for representations of a finite group G, the base field of choice is the field C of complex numbers; such representations are called complex representations of G. One encounters a very different flavor of representation theory when the characteristic of k divides the order of G; representations of this kind are referred to as modular representations of G. Our main focus in this chapter will be on non-modular representations of finite groups. Throughout, k denotes an arbitrary field and G is a group, generally in multiplicative notation. All further hypotheses will be explicitly spelled out when they are needed.

3.1 Group Algebras and Representations of Groups: Generalities This section lays the foundations of the theory of group representations by placing it in the framework of representations of algebras; this is achieved by means of the group algebra kG of G over k. 3.1.1 Group Algebras As a k-vector space, the group algebra of the group G over k is a k-vector space kG of all formal k-linear combinations of the elements of G; see Example A.5. Thus,

112

3 Groups and Group Algebras

P elements of kG can be uniquely written in the form x∈G λx x with λx ∈ k almost all 0. The (multiplicative) group operation of G gives rise to a multiplication for kG: X  X   X X X def λx x µy y = λx µy xy = λx µy z (3.1) x∈G

y∈G

x,y∈G

z∈G

x,y∈G xy=z

It is a routine matter to check that this yields an associative k-algebra structure on kG, with unit map k → kG is given by λ 7→ λ1G , where 1G is the identity element of the group G. Note that the basis elements x ∈ G are units (i.e., invertible elements) of the algebra kG. The group algebra kG is often denoted by k[G] in the literature, and we will use this notation as well, especially in cases where the notation for the group in question is sufficiently involved. Universal Property Despite being simple and natural, it may not be immediately clear why the above definition should be worth our attention. The principal reason is provided by the following “universal property” of the group algebra. For any k-algebra A, let A× denote its group of units. Then there is a natural bijection × HomAlgk (kG, A) ∼ = HomGroups (G, A )

(3.2)

The bijection is given by sending an algebra map f : kG → A to its restriction f |G to the basis G ⊆ kG as in (A.4). Observe that f |G is indeed a group homomorphism G → A× . Conversely, if G → A× is any group homomorphism, then its unique k-linear extension from G to kG is in fact a k-algebra map. Functoriality Associating to a given k-algebra A its group of units, A× , is a “functorial” process: Any algebra map A → B restricts to a group homomorphism A× → B × . The usual requirements on functors with respect to identities and composites are clearly satisfied; so we obtain a functor ·

×

: Algk → Groups

Similar things can be said for the rule that associates to a given group G its group algebra kG. Indeed, we have already observed above that every group G is a subgroup of the group of units kG× . Thus, if f : G → H is a group homomorphism, then the composite of f with the inclusion H ,→ kH × is a group homomorphism G → kH × . By (3.2) there is a unique algebra homomorphism kf : kG → kH such that the following diagram commutes:

3.1 Group Algebras and Representations of Groups: Generalities

113

f

G

H

∃! kf

kG

kH

It is straightforward to ascertain that k · respects identity maps and composites as is required for a functor, and hence we do have a functor k · : Groups → Algk Finally, it is routine to verify that the bijection (3.2) is functorial in both G and A; so the functor k · is left adjoint to the unit group functor · × . First examples Having addressed the basic formal aspects of group algebras in general, let us now look at two explicit examples of group algebras and describe their structure as algebras. Example 3.1 (The group algebra of a lattice). Free abelian groups of finite rank are often referred to as lattices. While it is desirable to keep the natural additive notation of Zn , the group law of Zn becomes multiplication in the group algebra kZn . In order to resolve this conflict, we will denote an element m ∈ Zn by xm when thinking of it as an element of the group algebra kZn . This results in the rule 0

xm+m = xm xm

0

(m, m0 ∈ Zn )

which governs the multiplication of kZn . Fixing a Z-basis {ei }n1 of Zn and putting xi = xei , each xm takes the form m

m

n xm = x1 1 x2 2 . . . xm n

with unique mi ∈ Z. Thus, the group algebra kZn is isomorphic to a Laurent polynomial algebra over k, ±1 ±1 ±1 kZn ∼ = k[x1 , x2 , . . . , xn ]

and the elements xm ∈ kZn are thought of as monomials. Replacing the group Zn by the submonoid Zn+ , we obtain an isomorphism of the monoid algebra kZn+ with the ordinary polynomial algebra k[x1 , x2 , . . . , xn ]. Example 3.2 (Group algebras of finite abelian groups). Now let G be finite abelian. Then G∼ = Cn1 × Cn2 × · · · × Cnt for suitable positive integers ni , where Cn denotes the cyclic group of order n. Sending a fixed generator of Cn to the variable x gives an isomorphism of algen bras kCn ∼ = k[x]/(x − 1). Moreover, the above group isomorphism for G yields an algebra isomorphism kG ∼ = kCn1 ⊗ kCn2 ⊗ · · · ⊗ kCnt ; see Exercise 3.1. Therefore,

114

3 Groups and Group Algebras

kG ∼ =

t O

k[x]/(xni − 1)

i=1

By field theory, this algebra is a direct product of fields if and only if char k does not divide any of the integers ni or, equivalently, char k - |G|. This is a very special case of Maschke’s Theorem (§3.4.1) which will be proved later in this chapter. Some variants
P
P The definition of the product x∈G λx x y∈G µy y in (3.1) also makes sense with any ring R in place of k, resulting in the group ring of RG. The case R = Z will play some role later in Section 10.3. We can also use an arbitrary monoid Γ instead of a group and obtain the monoid algebra kΓ or the monoid ring RΓ in this (I) way. For example, let I be any set and let Γ be the (additive) monoid Z+ of all finitely supported functions I → Z+ with pointwise addition: (m + n)i = mi + ni . Then one sees exactly as in Example 3.1 above that the resulting monoid ring RΓ is isomorphic to the polynomial ring R[xi | i ∈ I]. Finally, the product (3.1) also makes sense when possibly infinitely many λx or µy are nonzero, provided the monoid Γ satisfies the following condition: For all z ∈ Γ , the set {(x, y) ∈ Γ × Γ | xy = z} is finite. In this case, (3.1) defines a multiplication on the R-module RΓ of all functions Γ → R, not just on the submodule RΓ = R(Γ ) of all finitely supported functions. The (I) resulting ring is called the total monoid ring of Γ over R. For Γ = Z+ the condition above is easily seen to be satisfied, and we obtain the ring of formal power series RJxi | i ∈ IK in the commuting variables xi over R in this way. 3.1.2 Representations of Groups and Group Algebras Recall that a representation of the group G over k, by definition, is a group homomorphism G → GL(V ) ×

for some k-vector space V . Noting that GL(V ) = Endk (V ) , the adjoint functor relation (3.2) gives a natural bijection HomAlgk (kG, Endk (V )) ∼ = HomGroups (G, GL(V )) Therefore, representations of G over k are in natural 1-1 correspondence with representations of the group algebra kG: representations of kG

≡

representations of G over k

3.1 Group Algebras and Representations of Groups: Generalities

115

This makes the material of Chapter 1 available for the treatment of group representations. In particular, we may view the representations of G over k as a category that is equivalent to Rep kG or kG Mod, and we may speak of homomorphisms and equivalence of group representations as well as of irreducibility, composition series etc. The base field We mention that, in the context of group representations, the base field k is often understood and is notationally suppressed in the literature. Thus, for example, HomkG (V, W ) is often written as HomG (V, W ). Generally, we will use the former notation, which acknowledges the base field, except in Chapter 4 when considering the symmetric groups. We will however say that k is a splitting field for G rather than k is a splitting field for kG. Recall from §1.2.5 that this means that D(S) = k for all S ∈ Irrfin kG. Thus, an algebraically closed field k is a splitting field for any group G by Schur’s Lemma. Much less than algebraic closure is required for finite groups; see Exercise 3.4 and also Corollary 4.16. 3.1.3 Changing the Group We have seen that each group homomorphism H → G lifts uniquely to an algebra map kH → kG. Therefore, as discussed in §1.2.2 for arbitrary algebra homomorphisms, we have the restriction functor ReskG kH from kG-representations to kHrepresentations. In group representation theory, the following alternative notations are frequently used: G G kG · ↓H = · ↓H = ResH = ReskH : Rep kG −→ Rep kH

In the other direction, we have the induction and conduction functors. In the context of finite groups, which will be our main focus in this chapter, we may concentrate on induction – the reason for this is Proposition 3.3(b) below – for which the following notations are in use: ·↑

G

G kG = · ↑G H = IndH = IndkH : Rep kH −→ Rep kG

The following proposition states some facts that do not hold for general algebras. Notably, part (b) states that if H is a subgroup of G with finite index [G : H], then kG the functor CoindkG kH is isomorphic to the functor IndkH in the sense of Appendix A; thus, the embedding kH ,→ kG is a Frobenius extension (Exercise 2.22). We let G/H denote the collection of all left cosets gH (g ∈ G) and also, by a slight abuse of notation, any representative set of these cosets. Proposition 3.3. Let G be a group and let H be a subgroup of G. (a) Any kH-representation W is isomorphic to a subrepresentation W 0 of W↑G↓H . Moreover, W ↑G is the direct sum of the translates g.W 0 with g ∈ G/H. In particular, dimk W↑G = [G : H] dimk W

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3 Groups and Group Algebras

(b) If [G : H] < ∞, then, for each W in Rep kH, there is a natural isomorphism of kG-representations kG ∼ IndkG kH W = CoindkH W Proof. (a) The crucial observation is that kG is free as a right kH-module, a basis being provided by any fixed transversal for G/H; so elements of kG can be uniquely written as finite sums X a= gag g∈G/H

with ag ∈ kH that are almost all 0 (Exercise 3.2). Therefore, elements of W ↑G = kG ⊗kH W have the form X g ⊗ wg g∈G/H

with unique wg ∈ W . The map W → W ↑G↓H that is given by w 7→ 1 ⊗ w is a morphism in Rep kH and the foregoing implies that it is injective, since we may assume our chosen transversal to contain 1. Letting W 0 denote the image of this map, the translate g.W 0 consists of the elements g ⊗ w with w ∈ W . Thus, the above normal form of elements of W↑G also implies the remaining assertions of (a). (b) Consider the following projection of kG onto kH: kH

P

x∈G

(3.3)

∈

kG ∈

πH :

P

λx x

x∈H

λx x

Thus, πH is the identity on kH. Moreover, πH is a (kH, kH)-bimodule map – this is easy to see – and the following identity holds for every a ∈ kG: X X a= πH (ag)g −1 = gπH (g −1 a) (3.4) g∈G/H

g∈G/H

By linearity, it suffices to check this identity for a ∈ G, in which case it is immediate. The map πH leads to the following map of kH-representations: ∗

CoindkH kH W

w

b 7→ b.w

πH

CoindkG kH W↓H ∈

∼

∈

W ∈

φ:




a 7→ πH (a).w




By Proposition 1.9(a), the map φ has a unique extension to a kG-module map kG Φ : IndkG kH W → CoindkH W with Φ(a ⊗ w) = a.φ(w) for a ∈ kG and w ∈ W . In kG the opposite direction, define Ψ : CoindkG kH W → IndkH W by X Ψ (f ) = g ⊗ f (g −1 ) g∈G/H

Using (3.4), one verifies without difficulty that Ψ is an inverse map for Φ.

t u

3.1 Group Algebras and Representations of Groups: Generalities

117

Adjointness relations Let G be a group and let H be a subgroup of G. Then the adjointness relations of Proposition 1.9 can be stated as follows. For any W in Rep kH and V in Rep kG, we have a natural k-linear isomorphisms HomkG (W↑G , V ) ∼ = HomkH (W, V ↓H )

(3.5)

and, when [G : H] < ∞, HomkG (V, W↑G ) ∼ = HomkH (V ↓H , W )

(3.6)

The latter isomorphism does ofPcourse use Proposition 3.3; it sends a given f ∈ HomkH (V ↓H , W ) to the map g∈G/H g ⊗ f (g −1 · ) ∈ HomkG (V, W ↑G ). Both (3.5) and (3.6) will be referred to as Frobenius reciprocity isomorphisms. We conclude our first foray into the categorical aspects of group representations by giving some down-to-earth applications to irreducible representations. The argument for (b) was already used to similar effect in Exercise 1.21 in the more general context of cofinite subalgebras. Corollary 3.4. Let G be a group and let H be a subgroup of G. Then: (a) Every W in Irr kH is a subrepresentation of V ↓H for some V in Irr kG. Thus, any upper bound for the degrees of all irreducible representations of kG is also an upper bound for the degrees of the irreducible representations of kH. (b) Assume that [G : H] < ∞. Then every V in Irr kG is a subrepresentation of W↑G for some W ∈ Irr kH. Consequently, if the degrees of all irreducible representations of kH are bounded above by d, then all irreducible representations of kG have degrees at most [G : H]d. Proof. (a) We know by Proposition 3.3(a) that W embeds into W ↑G↓H and that W ↑G is a cyclic kG-module, because W is cyclic. Hence there is an epimorphism W ↑G  V for some V ∈ Irr kG (Exercise 1.3(a)). By (3.5), this epimorphism corresponds to a nonzero map of kH-representations W → V ↓H , which must be injective by irreducibility of W . This proves the first assertion of (a). The statement about degree bounds is clear. (b) The restriction V ↓H is finitely generated as kH-module, because V is cyclic and kG is finitely generated as left kH-module by virtue of our hypothesis on [G : H]. Therefore, there is an epimorphism V ↓H  W for some W ∈ Irr kH, and this corresponds to a nonzero map V → W ↑G by (3.6). By irreducibility of V , the latter map must be injective, proving the first assertion of (b). The degree bound is a consequence of the dimension formula in Proposition 3.3(a). t u

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3 Groups and Group Algebras

3.1.4 Characters of Finite-Dimensional Group Representations Let G be an arbitrary group. For V in Repfin kG, we have the associated character ∗

χV ∈ (kG)trace ⊆ (kG)∗
∗ ∗ where (kG)trace ∼ = kG/[kG, kG] is the space of all trace forms on kG as in (1.54). By (A.4), linear forms in (kG)∗ can be identified with functions G → k. Relations with conjugacy classes The group G acts on itself by conjugation, G×G

G

∈

∈

(3.7)

x

(x, y)

y := xyx−1

and the orbits in G under this action are called the conjugacy classes of G; the Gconjugacy class of x ∈ G will be denoted by G

x

Using bilinearity of the Lie commutator [ · , · ], one computes X X X [kG, kG] = k(xy − yx) = k(xy − y (xy)) = k(x − y x) x,y∈G

x,y∈G

x,y∈G ∗

Thus, a linear form φ ∈ (kG)∗ belongs to the subspace of trace forms (kG)trace if and only if φ is constant on the conjugacy classes of G. A function G → k that is constant on all conjugacy classes of G is called a k-valued class function on G. We summarize our discussion in the following diagram of k-linear isomorphisms:

∼



functions G → k

⊆ ∗

(kG)trace

(3.8)

⊆

(kG)∗

∼

def

cf k (G) =



class functions G → k



We will usually think of characters of kG-representations as class functions on G. Proposition 3.5. Assume that the group G has only finitely many conjugacy classes. Then  dimk cf k (G) = dimk kG/[kG, kG] = # conjugacy classes of G  In particular, # Irrfin kG ≤ # conjugacy classes of G .

3.1 Group Algebras and Representations of Groups: Generalities

119

Proof. In general, letting C = C (G) denote the set of conjugacy classes of G, we have a k-linear isomorphism C cf k (G) ∼ =k where kC denotes the vector space of all functions C → k. If C is finite, then a k-basis of cf k (G) is given by the functions δC : G → k (C ∈ C ) that are defined by δC (x) = 0 if x ∈ G \ C and δC (x) = 1 for all x ∈ C. This shows that dimk cf k (G) = #C . The equality dimk cf k (G) = dimk kG/[kG, kG] follows from (3.8), since finite-dimensional vector spaces are isomorphic to their duals. Finally, # Irrfin kG ≤ dimk C(kG) by Theorem 1.41. Since C(kG) is a subspace of Tr(kG)∗ ∼ t u = cf k (G), the bound for # Irrfin kG follows. There are infinite groups with finitely many conjugacy classes; in fact, every torsion free group embeds into a group with exactly two conjugacy classes [155, Exercise 11.78]. However, we will apply Proposition 3.5 to finite groups G only. In this case, the foregoing allows us to identify the space of class functions cf k (G) with Tr kG = kG/[kG, kG] by means of the isomorphism

∼

cf k (G)

∈ P

x∈G

(3.9)

∈

Tr kG

P

λx x + [kG, kG]

x∈C

λx

 C∈C (G)

Character tables Important representation theoretic information for a given finite group G is encoded in the character table of G; this is the matrix whose (i, j)-entry is χi (gj ), where {χi } are the irreducible characters of kG in some order, usually starting with χ1 = 1, and {gj } is a set of representatives for the conjugacy classes of G, generally with g1 = 1. Thus, the first column of the character table gives the degrees of the various irreducible representations, viewed in k. Usually, at least the sizes of the various conjugacy classes of G are also indicated in the character table and other information may be included as well. gj

classes sizes

1 1

... ...

1 .. .

1 .. .

...

1 .. .

...

...

χi (gj )

...

χi = χSi (dimk Si )1k .. .

.. .

G

| gj |

.. .

... ...

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3 Groups and Group Algebras

Conjugacy classes of p-regular elements We conclude this section with a result of Brauer [27] on representations of a finite group G in characteristic p > 0. An element g ∈ G is called p-regular if the order of g is not divisible by p. Because conjugate elements have the same order, we may also speak of p-regular conjugacy classes of G. Finally, any g ∈ G can be uniquely written as g = xy = yx where x ∈ G is p-regular and the order of y is a power of p. We will write x = gp0 and refer to gp0 as the p-regular part of g. Theorem 3.6 (Brauer). Let G be a finite group and let k be splitting field for G with ∗ char k = p > 0. Then the isomorphism (kG)trace ∼ = cf k (G) of (3.8) restricts to an isomorphism  C(kG) ∼ = f ∈ cf k (G) | f (g) = f (gp0 ) for all g ∈ G  In particular, # Irr kG = # p-regular conjugacy classes of G . Proof. We will prove the isomorphism for C(kG) and show that the dimension of this space is equal to the number of p-regular conjugacy classes of G. In view of Theorem 1.41, this will imply the formula for # Irr kG. By Proposition 1.40, we know that a linear form φ ∈ (kG)∗ belongs to C(kG) if and only if φ vanishes on  T = T (kG) = a ∈ kG | aq ∈ [kG, kG] for some q = pn , n ∈ Z+ Recall that T is a k-subspace of kG containing the commutator space [kG, kG] (Exercise 1.52). Fixing a set {xi } of representatives for the p-regular conjugacy classes of G, it suffices to show that {xi + T } is a k-basis of kG/T . Write a given g ∈ G as g = xy = yx with x = gp0 and y q = 1 for some q = pn . Then (g − x)q = g q − xq = 0; so g ≡ x mod T . Inasmuch as x ≡ xi mod T for some i, it follows that {xi +T } generates the k-vector space kG/T . In order to prove P linear independence, assume that i λi xi ∈ T with λi ∈ k. We need to show that all λi = 0. For this, recall from Exercise 1.52 that the commutator space [kG, kG] P is stable under the pth power map. Thus, ( i λi xi )q ∈ [kG, kG] for all sufficiently large q = pn . Writing |G| = pt m with (p, m) = 1 and choosing a large n so that q ≡ 1 mod m, we also have xqi = xi for all i. Since the pth -power map yields an additive endomorphism of kG/[kG, kG] by Exercise 1.52, we obtain X q X X q 0≡ λ i xi ≡ λqi xqi = λi xi mod [kG, kG] i

i

i

Finally, non-conjugate elements of G are linearly independent modulo [kG, kG] by Proposition 3.5, whence λqi = 0 for all i and so λi = 0 as desired. t u The result remains true for char k = 0, with the understanding that all conjugacy classes of a finite group G are 0-regular. Indeed, by Proposition 3.5, we already know that # Irr kG is bounded above by the number of conjugacy classes of G, and we shall see in Corollary 3.20 that equality holds if k is a splitting field for G.

3.1 Group Algebras and Representations of Groups: Generalities

121

3.1.5 Finite Group Algebras as Symmetric Algebras We mention here in passing that group algebras of finite groups fit into the context of symmetric algebras (§2.2.5). As we shall see later (§3.7.1), certain properties of finite group algebras are in fact most conveniently derived in this more general ring theoretic setting. In detail, for any group algebra kG, the map π1 in (3.3) is a trace form which we shall now denote by λ as in Section 2.2. Thus, using h · , · i for evaluation of linear forms as we did earlier, P hλ, g∈G αg gi = α1 (3.10) P If 0 6= a = g∈G αg g ∈ kG, then hλ, ax−1 i = αx 6= 0 for some x ∈ G. Thus, if the group G is finite, then λ is a Frobenius form for kG. Since hλ, gg 0 i = δg0 ,g−1 for g, g 0 ∈ G, the Casimir element and Casimir trace are given by X X (3.11) gag −1 and γλ (a) = g ⊗ g −1 cλ = g∈G

g∈G

In particular, γλ (1) = |G|1. Thus, if char k - |G|, then 1 = γλ (|G|−1 1) belongs to the Higman ideal Γ (kG) and so kG is semisimple by Theorem 2.24 or Proposition 2.18. The converse is also true and is in fact easier. Later in this chapter, we will give a direct proof of both directions that is independent of the material on symmetric algebras; see Maschke’s Theorem (§3.4.1).

Exercises 3.1 (Some group algebra isomorphisms). Establish the following k-algebra isomorphisms, for arbitrary groups G and H. (a) k[G × H] ∼ = kG ⊗ kH. op op op (b) k[Gop ] ∼ = (kG) . Here G is the opposite group: G = G as sets, but with new group operation ∗ given by x ∗ y = yx. (c) K ⊗ kG ∼ = KG for any field extension K/k. More generally, K can be any k-algebra here. 3.2 (Freeness over subgroup algebras and applications to induced representations). Let G be an arbitrary group and let H be a subgroup of G. (a) Show that kG is free as a left (and right) module over kH by multiplication: any set of right (resp., left) coset representatives for H in G provides a basis. (b) Conclude from (a) and Exercise 1.16 that, for any W in Rep kH, we have Ker(W ↑G ) = {a ∈ kG | a kG ⊆ kG Ker W }, the largest ideal of kG that is contained in the left ideal kG Ker W . (c) Let W be a finite-dimensional representation of kH. Use Proposition 3.3 to show that the character of the induced representation W↑G is given by

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3 Groups and Group Algebras

χW↑G (x) =

X

χW (g −1 xg)

g∈G/H −1 g xg∈H

(d) Use Proposition 3.3 to show that the induction functor ↑G is exact: for any short exact sequence 0 → W 0 → W → W 00 → 0 in Rep kH, the sequence 0 → W 0↑G → W↑G → W 00↑G → 0 is exact in Rep kG. 3.3 (An irreducibility criterion). Let G be aL group and let H be a subgroup of G. Assume that V ∈ Rep kG is such that V ↓H = i∈I Wi for pairwise non-isomorphic irreducible subrepresentations Wi such that kG.Wi = V for all i. Show that V is irreducible. 3.4 (Splitting fields in positive characteristic). Let G be a finite group and let e = exp G denote the exponent of G, that is, the least common multiple of the orders of all elements of G. Let k be a field with char k = p > 0 and assume that k contains µe , the group of eth roots of unity in some fixed algebraic closure k of k. Show that k is a splitting field for G. [Use Exercise 1.51.] 1 3.5 (Hattori’s Lemma). Let G be a finite group and let P ∈ kG proj be a finitely generated projective kG-module. Using the isomorphism (3.9), we may view the ∼ Hattori-Stallings rank as a function rank : K0 (kG) −→ Tr(kG) −→ cf k (G) ; see (2.11). Use Proposition 2.9 to establish the formula χP (g) = |CG (g)| rank(P )(g −1 )

(g ∈ G)

3.5

3.2 First Examples 3.2.1 Finite Abelian Groups Let G be a finite abelian group. Then the group algebra kG is a finite-dimensional commutative algebra, and so we know by (1.37) that there is a bijection

∈

∈

MaxSpec kG ∼ Irr kG

P

kG/P

The Schur division algebra of the irreducible representation S = kG/P is given by D(S) = EndkG (kG/P ) ∼ = kG/P . Let e = exp G denote the exponent of G (i.e., 1

The fact stated in this exercise is true in characteristic 0 as well by another result of Brauer. For a proof, see [88, Theorem (10.3)] for example; an easy special case is (3.12) below.

3.2 First Examples

123

the smallest positive integer e such that xe = 1 for all x ∈ G), and let µe denote the group of eth roots of unity in some fixed algebraic closure k of k: def

µe = {λ ∈ k | λe = 1} Consider the subfield K = k(µe ) ⊆ k and the group algebra KG ∼ = K ⊗ kG. Every Q ∈ MaxSpec KG satisfies KG/Q ∼ = K, because the images of all elements of G in the field KG/Q are eth roots of unity, and hence they all belong to K. Thus, k(µe ) is a splitting field for G.

(3.12)

For each P ∈ MaxSpec kG, the field kG/P embeds into the field K; so there is a k-algebra map f : kG → K with Ker f = P . Conversely, if f ∈ HomAlgk (kG, K), then Ker f ∈ MaxSpec kG. Put Γ = Gal(K/k) and consider the Γ -action on HomAlgk (kG, K) that is given by γ.f = γ∗ (f ) = γ ◦ f for γ ∈ Γ and f ∈ HomAlgk (kG, K). Then Ker f = Ker f 0 holds for f, f 0 ∈ HomAlgk (kG, K) if and only if f and f 0 belong to the same Γ -orbit. Thus, letting Γ \ HomAlgk (kG, K) de∼ note the set of Γ -orbits in HomAlgk (kG, K), there is a bijection MaxSpec kG ←→ Γ \ HomAlgk (kG, K). All this holds more generally in the setting of algebras; see Exercise 1.37. Identifying HomAlgk (kG, K) with HomGroups (G, K × ) by (3.2), we obtain a bijection ×

∼ Irr kG ←→ Gal(k(µe )/k)\ HomGroups (G, k(µe ) )

(3.13)

3.2.2 Degree-1 Representations Recall from (1.36) that, for any k-algebra A, the equivalence classes of degree-1 representations form a subset of Irr A that is in natural 1-1 correspondence with × HomAlgk (A, k). If A = kG, then HomAlgk (kG, k) ∼ = HomGroups (G, k ) by (3.2). Thus, we have a bijection n

equivalence classes of degree-1 representations of kG

∈

∼

∈

HomGroups (G, k× )

φ

kφ

o

⊆

Irr kG

Here kφ denotes the field k with G-action g.λ = φ(g)λ for g ∈ G and λ ∈ k as in (1.36). Note that φ = χkφ is the character of kφ . Occasionally, we will simply write φ in place of kφ . The group structure of k× endows HomGroups (G, k× ) with the structure of an abelian group: (φψ)(x) = φ(x)ψ(x). The identity element of this group is the socalled trivial representation:

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3 Groups and Group Algebras

∈

k×

∈

1: G x

1

(3.14)

Letting Gab = G/[G, G] denote the abelianization of G, we have a canonical bijection (Example A.4) ab × HomGroups (G, k× ) ∼ = HomGroups (G , k )

In order to further describe this group, we concentrate on the case where Gab is finite and we let e = exp(Gab ) denote the exponent of Gab . Then we have the following two extreme cases: • If char k - e and µe ⊆ k, then there is a (non-natural) isomorphism of groups ab HomGroups (Gab , k× ) ∼ = G (Exercise 3.6). Thus: Proposition 3.7. Assume that Gab is finite with exponent e. If char k - e and µe ⊆ k, then the number of non-equivalent degree-1 representations of G is equal to |Gab |. • If char k = p > 0 and e is a power of p, then µe = {1} and then HomGroups (Gab , k× ) = {1}, because µe = {1}. Thus, up to equivalence, the trivial representation is the only degree-1 representation of G. In fact, for finite p-groups we have the following important fact. Proposition 3.8. If char k = p > 0 and G is a finite p-group, then 1 is the only irreducible representation of kG up to equivalence. Proof. The case G = 1 being obvious, assume that G 6= 1 and let S ∈ Irr kG. Our hypotheses on k and G imply that S is finite-dimensional and 1 is the only eigenvalue of gS for all g ∈ G. Choosing 1 6= g ∈ Z G, the 1-eigenspace of gS is a nonzero subrepresentation of S, and hence it must be equal to S. Thus, S is a representation of k[G/hgi], clearly irreducible, and so S ∼ t u = 1 by induction on the order of G. 3.2.3 The Dihedral Group D4 The dihedral group Dn is given by the presentation def

Dn = hx, y | xn = 1 = y 2 , xy = yxn−1 i

(3.15)

Geometrically, Dn can be described as the symmetry group of the regular n-gon in R2 . The order of Dn is 2n and Dn has the structure of a semidirect product: Dn = hxi o hyi ∼ = Cn o C2 ab Since x2 = [y, x] ∈ [Dn , Dn ], it is easy to see that Dnab ∼ = C2 for odd n, and Dn ∼ = C2 × C2 for even n.

3.2 First Examples

125

v2 Let us now focus on D4 and work over any base field k ab ∼ with char k 6= 2. Since D4 = C2 × C2 , we know that the degree-1 representations of D4 are given by the four choices x of φ±,± : x 7→ ±1, y 7→ ±1; so φ+,+ = 1 is the trivial v1 representation. Another representation arises from the realization of D4 as the symmetry group of the square, with x acting as the counterclockwise rotation by π/2 and y as the y reflection across the vertical axis of symmetry; see the pic
0 −1 ture on the right. With respect to the indicated basis v , v , the matrix of x is 1 2 1 0
0 and y has matrix −1 0 1 . These matrices make sense in Mat2 (k) and they satisfy the defining relations (3.15) of D4 ; hence, they yield a representation of kD4 . Let us call this representation S. Since the two matrices for x and y have no common eigenvector, S is irreducible. Furthermore, D(S) = k: only the scalar matrices commute with the matrix of x and the matrix of y. Finally, it is easy to check that the matrices x and y generate the k-algebra Mat2 (k) – this also follows from Burnside’s Theorem (§1.4.5). To summarize, we have constructed five non-equivalent absolutely irreducible representations of kD4 : the four degree-1 representations φ±,± and S, of degree 2. Their kernels are distinct maximal ideals of kD4 , with factors isomorphic to k for all kD4 / Ker φ±,± ∼ = k and to kD4 / Ker S ∼ = Mat2 (k). As in (1.56), the Chinese Remainder Theorem yields an epimorphism of k-algebras kD4  k × k × k × k × Mat2 (k), which must be an isomorphism for dimension reasons. Thus: kD4 ∼ = k × k × k × k × Mat2 (k)

In particular, kD4 is split semisimple and Irr kD4 = {1, φ+,− , φ−,+ , φ−,− , S}. Note also that D4 has five conjugacy classes, with representatives 1, x2 , x, y and xy. We record the character table of kD4 in Table 3.1; all values in this table have to be interpreted in k. 2

classes sizes

1 1

x 1

x 2

y 2

xy 2

1

1

1

1

1

1

φ−,+

1

1

-1

1

-1

φ+,−

1

1

1

-1

-1

φ−,−

1

1

-1

-1

1

χS

2

-2

0

0

0

Table 3.1. Character table of D4 in k (char k 6= 2)

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3 Groups and Group Algebras

3.2.4 Some Representations of the Symmetric Group Sn Let Sn denote the group of all permutations of the set [n] = {1, 2, . . . , n} and assume that n ≥ 2. Then Snab = Sn /An ∼ = C2 , where An is the alternating group consisting of the even permutations in Sn . Thus, besides the trivial representation 1, there is only one other degree-1 representation, up to equivalence, and only if char k 6= 2: the so-called sign representation × sgn : Sn  Snab ∼ = {±1} ⊆ k

In order to find additional irreducible representation of Sn , we use the action of Sn on the set [n], which we will write as [n] = {b1 , b2 , . . . , bn } so as to not confuse its elements with scalars from k. Let Mn = k[n] denote the k-vector space with basis [n]. The standard permutation representation of Sn is defined by def

Mn =

n M

kbi

with s.bi = bs(i)

(s ∈ Sn )

(3.16)

i=1

∼ GLn (k) that is provided by the basis of In terms of the isomorphism GL(Mn ) = [n] of Mn , the image of the homomorphisms Sn → GLn (k) consists exactly of the permutation matrices, having exactly one entry 1 in each row and column with all other entries being 0. Note that Mn is not irreducible: the 1-dimensional subspace spanned by the vecP tor i bi ∈ Mn is a proper subrepresentation of Mn that is equivalent to the trivial representation 1. Also, the map k=1

∈

∈

π : Mn P

i

λi bi

P

i

(3.17)

λi

is easily seen to be an epimorphism of representations. Therefore, we obtain a representation of degree n − 1 by putting def

Vn−1 = Ker π This is called the standard representation of Sn . It is not hard to show that Vn−1 is irreducible if and only if either n = 2 or n > 2 and char k - n. Furthermore, one always has EndkSn (Vn−1 ) = k; see Exercise 3.8 for all this. Consequently, if n > 2 and char k - n, then Burnside’s Theorem (§1.4.5) provides us with an epimorphism of algebras kSn  BiComkSn (Vn−1 ) ∼ = Matn−1 (k). Example 3.9 (The structure of kS3 ). Assume that char k 6= 2, 3. (See Exercise 3.36 for characteristics 2 and 3.) Then we have three non-equivalent irreducible representations for S3 over k: 1, sgn and V2 . Their kernels are three distinct maximal ideals

3.2 First Examples

127

of kS3 , with factors k, k and Mat2 (k), respectively. Exactly as for kD4 above, we obtain an isomorphism of k-algebras, kS3 ∼ = k × k × Mat2 (k) Thus, kS3 is split semisimple and Irr kS3 = {1, sgn, V2 }. Note also that S3 has three conjugacy classes, with representatives (1), (1 2) and (1 2 3). With respect to the basis {b1 − b2 , b2 − b3 } of V2 , the operators (1 2)V2 and (1 2 3)V2 have matrices

0 −1 −1 1 0 1 and 1 −1 , respectively. In particular, we obtain the character table of kS3 as recorded in Table 3.2. classes sizes

(1) 1

(1 2) 3

(1 2 3) 2

1

1

1

1

sgn

1

-1

1

χV 2

2

0

-1

Table 3.2. Character table of S3 in k (char k 6= 2, 3)

v2 We remark that S3 is isomorphic to the dihedral group D3 , the group of symmetries of a unilateral triangle, by sending (1 2) to the reflection across the vertical line of sym(1 2 3) metry and (1 2 3) to counterclockwise rotation by 2π/3 as v1 in the picture on the right. If k = R, then we may regard 2 V2 ∼ as the Euclidean plane. Using the basis consisting =R √ (1 2) of v1 = 3(b1 − b2 ) and v2 = b1 + b2 − 2b3 , the matri- 
cos 2π/3 − sin 2π/3 −1 0 ces of (1 2)V2 and (1 2 3)V2 are 0 1 and sin 2π/3 cos 2π/3 , respectively. Thus, V2 also arises from the realization of S3 as the group of symmetries of a unilateral triangle.

3.2.5 Permutation Representations

∈

∈

Returning to the case of an arbitrary group G, let us now consider a G-set X, that is, a set with a G-action G×X X (g, x)

g.x

satisfying the usual axioms: 1.x = x and g.(g 0.x) = (gg 0 ).x for all g, g 0 ∈ G and x ∈ X. We will usually write G X to indicate a G-action on the set X. Such an action extends uniquely to an action of G by k-linear automorphisms on the vector

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space kX of all formal k-linear combinations of the elements of X (Example A.5), thereby giving rise to a representation ρX : G → GL(kX) Representations V ∈ Rep kG that are equivalent to a representation of this form are called permutation representations of G; they are characterized by the fact that the action of G on V stabilizes some k-basis of V . If the set X is finite, then we can consider the character χkX : G → k; it is evidently given by χkX (g) = # FixX (g)1k

(g ∈ G)

(3.18)

where FixX (g) = {x ∈ X | gx = x} denotes the set of all fixed points of g in X. Examples 3.10. (a) If |X| = 1 then kX ∼ = 1. (b) Taking X = G, with G acting on itself by left multiplication, we obtain the regular representation ρG = ρreg of kG; it will also be denoted by (kG)reg as in Example 1.6. If G is finite, then the regular character of kG is given by ( |G|1k for g = 1 χreg (g) = 0 otherwise or P χreg ( g∈G αg g) = |G|α1 Viewing kG as a symmetric algebra as in §3.1.5, this formula is identical to (2.28). (c) We can also let G act on itself by conjugation: g.x = g x = gxg −1 for g, x ∈ G. The resulting permutation representation is called the adjoint representation of kG; it will be denoted by (kG)ad . If G is finite, then we may consider the character of this representation: χad (g) = |CG (g)|1k where CG (g) denotes the centralizer of g in G. (d) With G = Sn acting as usual on X = [n], we recover the standard permutation representation Mn of Sn . Recall from (3.17) that there is a short exact sequence of Sn -representations 0 → Vn−1 → Mn → 1 → 0. Thus, (3.18) in conjunction with Lemma 1.39 gives χMn (s) = # Fix(s)1k χVn−1 (s) = # Fix(s)1k − 1k for any s ∈ Sn , were Fix(s) = Fix[n] (s) is the number of 1-cycles in the disjoint cycle decomposition of s.

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Exercises 3.6 (Dual group). Let G be a finite abelian group of exponent e and assume that × char k - e. Show that there is a group isomorphism G ∼ = HomGroups (G, k(µe ) ). 3.7 (Splitting fields). Let G be an arbitrary group. Recall that k is said to be a splitting field for G if D(S) = k for all S ∈ Irrfin kG. Show: (a) If k is a splitting field for G, then k is also a splitting field for all homomorphic images of G. (b) Assume that k is a splitting field for G and that Gab is finite. Show that µe ⊆ k, where e = exp(Gab ). (c) Give an example showing that if k is a splitting field for G, then k need not be a splitting field for all subgroups of G. 3.8 (The standard representation of Sn ). Let Vn−1 (n ≥ 2) be the standard representation of the symmetric group Sn . Show: (a) Vn−1 is irreducible if and only if n = 2 or char k - n; (b) EndkSn (Vn−1 ) = k. 3.9 (The character table does not determine the group). Consider the real quaternions, H = R ⊕ Ri ⊕ Rj ⊕ Rk with i2 = j 2 = k 2 = ijk = −1, and the quaternion group Q8 = hi, ji = {±1, ±i, ±j, ±k} ≤ H× . Show that Q8 has the same character table over any field k with char k 6= 2 as the dihedral group D4 (Table 3.1), even though Q8  D4 .

3.3 More Structure Throughout this section, G denotes an arbitrary group unless mentioned otherwise. 3.3.1 Comultiplication, Counit, and Antipode In this subsection, we will apply the group algebra functor k ? : Groups → Algk to construct some k-algebra maps that add important structure to the algebra kG and its category of representations. The algebra kG, together with these new maps, becomes an example of a Hopf algebra. Chapters 9 and 12 of this book will be devoted to Hopf algebras and their representations. Comultiplication and counit The diagonal group homomorphism G → G × G, x 7→ (x, x), together with the ∼ isomorphism k[G × G] −→ kG ⊗ kG (Exercise 3.1) yields the algebra map kG ⊗ kG

P

x

λx x

∈

kG ∈

∆:

P

x

λx (x ⊗ x)

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3 Groups and Group Algebras

This map is called the comultiplication of kG. Similarly, the trivial group homomorphism G → {1} gives rise to the algebra map k ∈

kG ∈

ε:

P

x

P

λx x

x

λx

The map ε is referred to as the augmentation map or counit of kG; it is the same as the trivial representation 1 in (3.14), but the notation ε is standard in the context of Hopf algebras. We can also think of ε as a homomorphism of kG-representations ε : (kG)reg → 1. The kernel of ε is called the augmentation ideal of kG; we will use the notation def (kG)+ = Ker ε (3.19) As a k-subspace of kG, the augmentation ideal (kG)+ is generated by the elements g − 1 with g ∈ G. The nomenclature “comultiplication” and “counit” derives from the fact that these maps fit into commutative diagrams resembling the diagrams (1.1) for the multiplication and unit maps of kG, except that all arrows point in the opposite direction: kG ⊗ kG ⊗ kG

∆ ⊗ Id

kG ⊗ kG kG ⊗ kG

Id ⊗∆

ε ⊗ Id

and

∆

Id ⊗ε

k ⊗ kG

kG ⊗ kG

∆

kG ⊗ k

∆

∼

(3.20)

∼

kG kG

Both diagrams are manifestly commutative. The property of ∆ that is expressed by the diagram on the left is called coassociativity. Another notable property of the comultiplication ∆ is its cocommutativity: Letting τ : kG ⊗ kG → kG ⊗ kG denote the map given by τ (a ⊗ b) = b ⊗ a, we have ∆=τ ◦∆

(3.21)

Again, this concept is dual to the usual concept of commutativity: An algebra A with multiplication m : A ⊗ A → A is commutative if and only if m = m ◦ τ . Antipode ∼ Inversion gives a group isomorphism G −→ Gop , x 7→ x−1 . Here Gop denotes the op opposite group as in Exercise 3.1: G = G as sets, but with new group operation ∗ given by x ∗ y = yx. We obtain a k-linear map

kG

P

x

λx x

(3.22)

∈

∈

S : kG P

x

λx x

−1

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satisfying S(ab) = S(b)S(a) for all a, b ∈ kG. Note also that S2 = Id. The map S is called the standard involution or the antipode of kG. We can also think of S as an op ∼ isomorphism of algebras S : kG −→ k[Gop ] ∼ = (kG) . 3.3.2 Invariants Let V be any representation of kG. The k-subspace of G-invariants in V is defined by def

VG =

 v ∈ V | g.v = v for all g ∈ G

Evidently V G can also be described as the common kernel of the operators in G (kG)+ = V (1) , the 1-homogeneous compoV , and yet another description is V nent of V :   V G = v ∈ V | (kG)+ .v = 0 = v ∈ V | a.v = ε(a)v for all a ∈ kG More generally, if kφ is any degree-1 representation of G, given by a group homomorphism φ : G → k× , then the homogeneous component V (kφ ) will be written as Vφ as in Example 1.28: def

Vφ =

 v ∈ V | g.v = φ(g)v for all g ∈ G

Recall that the nonzero spaces Vφ are referred to as weight spaces; their elements are called weight vectors or semi-invariants. Invariants of permutation representions Let X be a G-set and let V = kX P be the associated permutation representation of kG as in §3.2.5. An element v = x∈X λx x ∈ kX belongs to (kX)G if and only if λg.x = λx for all x ∈ X and g ∈ G; in other words, the function X → k, x 7→ λx , is constant on all G-orbits G.x ⊆ X. Since λx = 0 for almost all x ∈ X, we conclude that if λx 6= 0 then x must belong to the following G-subset of X:  Xfin = x ∈ X | the orbit G.x is finite For each orbit O ∈ G\Xfin , we may define the orbit sum def

σO =

X

x

∈ (kX)G

x∈O

Denoting the commonP value of all λx with x ∈ O by λO , we can write the given v ∈ (kX)G as v = O∈G\Xfin λO σ O . Finally, since distinct orbits are disjoint,

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3 Groups and Group Algebras

the distinct orbit sums are linearly independent, and hence they yield a k-basis of (kX)G . To summarize, M (kX)G = k σO (3.23) O∈G\Xfin

Note also that k need not be a field in the foregoing. Example 3.11 (Invariants of the adjoint representation). The invariants of the adjoint representation from Example 3.10 are exactly the center of kG:  −1 (kG)G = a for all g ∈ G = Z (kG) ad = a ∈ kG | gag Moreover, the orbits of the conjugation action G G are the conjugacy classes of G. Thus, the orbit sums of the finite conjugacy classes of G — they are also called the class sums of G — form a k-basis of Z (kG) by (3.23). Invariants for finite groups: averaging Applying (3.23) to the regular representation (kG)reg of an arbitrary group G and noting that X = G consists of just one G-orbit, we obtain ( 0 if G is infinite G (kG)reg = (3.24) k σ G if G is finite P with σ G = g∈G g for finite G. Continuing to assume that G is finite and P considering an arbitrary representation V of kG, the operator V → V , v 7→ g∈G g.v, clearly has image in V G . If the order |G| is invertible in k, then all G-invariants in V are obtained in this way: Proposition 3.12. The counit ε is nonzero on (kG)G reg if and only if the group G is finite and char k - |G|. In this case, 1 1 X e= σG = g |G|

|G|

g∈G

2 is the unique element e ∈ (kG)G reg satisfying ε(e) = 1 or, equivalently, 0 6= e = e . Moreover, for every representation V of kG, the “averaging operator”

V

∈

∈

eV : V v

1 |G|

P

g∈G

g.v

is a projection of V onto V G . If V is finite-dimensional, then 1 X dimk V G · 1k = χV (e) = χV (g) |G|

g∈G

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Proof. If ε is nonzero on (kG)reg , then G must be finite by (3.24), and ε((kG)G reg ) = k|G|. The first assertion follows from this. Now assume that G is finite with char k G |G|. Then e ∈ (kG)G reg by (3.24) and e is the unique element of (kG)reg such that ε(e) = 1. It follows that eV ⊆ V G for every V and, for every v ∈ V G , we have ev = ε(e)v = v. Thus, V G = eV . In particular, with v = e ∈ (kG)G reg , we obtain 2 2 G e = e. Conversely, 0 6= e = e ∈ (kG)reg implies that ε(e) = 1. Finally, with respect to a k-basis of V = eV ⊕ (1 − e)V = V G ⊕ (1 − e)V that is the union of bases of V G and (1 − e)V , the matrix of the projection eV has the form   IdV G

 

0

0  0

Therefore, χV (e) = trace(eV ) = dimk V G · 1k , which completes the proof.

t u

The following well-known corollary is variously called Burnside’s Lemma or the Cauchy-Frobenius Lemma, the latter attribution being historically more correct. Corollary 3.13. If a finite group G acts on a finite set X, then the number of Gorbits in X is equal to the average number of fixed points of elements of G: 1 X # G\X = # FixX (g) |G|

g∈G

Proof. By (3.23) we know that dimQ (QX)G = # G\X while (3.18) tells us that the character of QX is given by χQX (g) = # FixX (g) for g ∈ G. The corollary therefore follows from the dimension formula for invariants in the proposition. t u 3.3.3 A Plethora of Representations The structure maps in §3.3.1 allow us to construct many new representations of kG from one or several given representations. This is sometimes referred to under the moniker “plethysm”, although this term originally had a somewhat more specific meaning; see Macdonald [121, I.8]. Constructions analogous to those to be considered below will be carried out later also in the context of Lie algebras and, more generally, Hopf algebras. We will then indulge in some more plethysm, and we will be able to refer to some of the detailed explanations that are given below. Homomorphisms If V and W are two representations of kG, then the k-vector space Homk (V, W ) can be made into a representation of kG by defining def

(g.f )(v) = g.f (g −1 .v)

(g ∈ G, v ∈ V, f ∈ Homk (V, W ))

(3.25)

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3 Groups and Group Algebras

Even though it is straightforward to verify directly that this rule does indeed define a representation, let us place it in a more conceptual framework. If V and W are representations of arbitrary algebras B and A, resp., then Homk (V, W ) becomes a representation of the algebra A ⊗ B op as in Example 1.3: def

(a ⊗ bop )f = aW ◦ f ◦ bV

(a ∈ A, b ∈ B, f ∈ Homk (V, W ))

Thus, we have an algebra map A ⊗ B op → Endk (Homk (V, W )). For A = B = kG, we also have (Id ⊗S) ◦ ∆ : kG → kG ⊗ kG → kG ⊗ (kG)op . The composite of these two algebra maps leads to (3.25). The bifunctor Homk for k-vector spaces (§B.3.2) restricts to a bifunctor Homk ( · , · ) : (Rep kG)op × Rep kG −→ Rep kG Here, we use op for the first variable, because Homk is contravariant in this variable while Homk is covariant in the second variable: for any V ∈ Rep kG and any morphism f : W → W 0 in Rep kG, we have Homk (f, V ) = f ∗ = · ◦ f : Homk (V, W 0 ) → Homk (V, W ) but Homk (V, f ) = f∗ = f ◦ · : Homk (V, W ) → Homk (V, W 0 ). It is readily verified that f ∗ and f∗ are indeed morphisms in Rep kG. Recall also that Homk is exact in either argument (§B.3.2). Evidently, g.f = f holds for all g ∈ G if and only if f (g.v) = g.f (v) for all g ∈ G and v ∈ V , and the latter condition in turn is equivalent to f (a.v) = a.f (v) for all a ∈ kG and v ∈ V . Thus, the G-invariants of Homk (V, W ) are exactly the homomorphism of representations V → W : Homk (V, W )G = HomkG (V, W )

(3.26)

∈

∈

Example 3.14. It is clear from the definitions that we have an isomorphism of representations Homk (1, V ) ∼ V f

f (1)

∼ This map restricts to an isomorphism HomkG (1, V ) −→ V G by (3.26).

Duality Taking W = 1 = kε in the preceding paragraph, the dual vector space V ∗ = Homk (V, k) becomes a representation of kG. By (3.25), the G-action on V ∗ is given by (g.f )(v) = f (g −1 .v)

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135

for g ∈ G, v ∈ V and f ∈ V ∗ or, equivalently, a.f = f ◦ S(a)V

(a ∈ kG, f ∈ V ∗ )

where S is the antipode (3.22). By our remarks about Homk , duality gives an exact contravariant functor ∗ · : Rep kG → Rep kG Lemma 3.15. Let V be a finite-dimensional representation of kG. Then the character of the dual representation V ∗ is given by χV ∗ (g) = χV (g −1 ) for g ∈ G. Proof. The action of a ∈ kG on V ∗ can be written in the above notation as aV ∗ = S(a)∗V where S(a)∗V is the transpose of the operator S(a)V as in §B.3.2. Since trace(S(a)V ) = trace(S(a)∗V ) by (B.24), we obtain χV ∗ (a) = χV (S(a)). The lemma follows, because S(g) = g −1 for g ∈ G. t u ∗ A representation V is called self-dual if V ∼ = V . Note that this forces V to be ∗ finite-dimensional, because otherwise dimk V > dimk V (§B.3.2). The lemma below shows that finite-dimensional permutation representations are self-dual; further self-dual representations can be constructed with the aid of Exercise 3.18.

Lemma 3.16. The permutation representation kX for a finite G-set X is self-dual. Proof. Let {δx }x∈X ∈ (kX)∗ be the dual basis for the basis X of kX; so δx (y) = δx,y 1k ∼ for x, y ∈ X. Then x 7→ δx defines a k-linear isomorphism δ : kX −→ (kX)∗ . We claim that this is in fact an isomorphism of representations, that is, δ(a.v) = a.δ(v) holds for all a ∈ kG and v ∈ kX. By linearity, we may assume that a = g ∈ G and v = x ∈ X. Then, for any y ∈ X,

δg.x (y) = δg.x,y 1k = δx,g−1 .y 1k = δx (g −1 .y) = (g.δx )(y) t u

and so δ(a.v) = a.δ(v) as desired. Tensor products

If V and W are given representations of kG, then the tensor product V ⊗W becomes a representation of kG via the “diagonal action” def

g.(v ⊗ w) = g.v ⊗ g.w

(g ∈ G, v ∈ V, w ∈ W )

(3.27)

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3 Groups and Group Algebras

The map τ (v ⊗ w) = w ⊗ v gives an isomorphism of representations V ⊗ W ∼ = W ⊗ V , and it is also clear that V ⊗ 1 ∼ = V . Finally, the G-action (3.27) clearly is compatible with the standard associativity isomorphism for tensor products; so the tensor product of kG-representations is associative. Again, let us place the action rule (3.27) in a more general context; this will have the added benefit of taking care of any well-definedness issues and the representation axioms. Recall from (1.51) that the outer tensor product of representations V ∈ Rep A and W ∈ Rep B, for arbitrary algebras A and B, is a representation of the algebra A⊗B: the algebra map A⊗B → Endk (V ⊗W ) is given by a⊗b 7→ aV ⊗bW . If A = B = kG, then we also have the comultiplication ∆ : kG → kG ⊗ kG. The composite of all these maps is an algebra map kG → Endk (V ⊗ W ) that gives the diagonal G-action on V ⊗ W in (3.27): gV ⊗W = gV ⊗ gW . Canonical isomorphisms The standard natural morphisms in Vectk discussed in Appendix B actually are morphisms in Rep kG; the reader will be asked in Exercises 3.17 and 3.20 to check this. Specifically, for U, V, W ∈ Rep kG, the Hom-⊗ adjunction isomorphism (B.14) is in fact an isomorphism in Rep kG: Homk (U ⊗ V, W ) ∼ = Homk (U, Homk (V, W ))

(3.28)

Similarly the canonical monomorphisms W ⊗V ∗ ,→ Homk (V, W ) and V ,→ V ∗∗ in (B.17) and (B.21) are morphisms in Rep kG, and so is the trace map Endk (V ) → k for a finite-dimensional V ∈ Rep kG when k is viewed as the trivial representation, k = 1. Thus, we have the following isomorphisms in Rep kG. If dimk V < ∞ or dimk W < ∞, then W ⊗V∗ ∼ = Homk (V, W )

(3.29)

∗ Homk (U ⊗ V, W ) ∼ = Homk (U, W ⊗ V )

(3.30)

and

for any U ∈ Rep kG. If V is finite-dimensional, then ∗∗ V ∼ =V

(3.31)

Lemma 3.17. Let V and W be finite-dimensional representations of kG. Then the characters of the representations V ⊗ W and Homk (V, W ) are given, for g ∈ G, by (a) χV ⊗W (g) = χV (g)χW (g) (b) χHomk (V,W ) (g) = χW (g)χV (g −1 ) Proof. Recall that gV ⊗W = gV ⊗ gW . Thus, (a) is a special case of formula (1.52). Finally, in view of the isomorphism (B.18), part (b) follows from Lemma 3.15 in conjunction with (a). t u

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137

3.3.4 Characters and Symmetric Polynomials Using the diagonal action (3.27), we inductively obtain diagonal G-actions on all tensor powers V ⊗k of a given kG-representation V , making each V ⊗k a representation of V ⊗0 = 1 is the trivial representation. Hence the tensor algebra LkG. Here, ⊗k TV = k≥0 V becomes a kG-representation as well. The actions of an element L ⊗k g ∈ G on V and on TV are given by the automorphisms gV⊗k and TgV = k gV⊗k as in §1.1.2. Similarly, the symmetric algebra Sym V and the exterior algebra ΛV , as well as their homogeneous components, become representations of kG, with g ∈ G acting by Sym gV and ΛgV , respectively. Assuming V to be finite-dimensional, the kG-representations V ⊗k , Symk V and Λk V are all finite-dimensional. In this section, we describe their characters. Generating functions Let V be a finite-dimensional vector space over k. Then any f ∈ Endk (V ) gives L ⊗k rise to the endomorphism Tf = of TV as in §1.1.2. We will conkf sider the generating function of the traces trace(f ⊗k ), that is, the power series P ⊗k k )t ∈ kJtK. Using the formula trace(f ⊗k ) = trace(f )k , which k≥0 trace(f follows from (B.25), we can rewrite this generating function as follows: X k≥0

trace(f ⊗k )tk =

1 1 − trace(f )t

(3.32)

P P Similarly, we have the series k≥0 trace(Symk f )tk and k≥0 trace(Λk f )tk for the endomorphisms Symf and Λf . The latter series is in fact a polynomial, because Λk V = 0 for k > dimk V . The following lemma, a one-variable version of the “MacMahon Master Theorem” [122, p. 97-98] from enumerative combinatorics, gives expressions for these generating functions analogous to (3.32). The lemma depends on some basic facts concerning symmetric polynomials. A good reference for this material is Macdonald’s monograph [121]. Lemma 3.18. Let V be a finite-dimensional k-vector space and let f ∈ Endk (V ). Then X 1 trace(Symk f )tk = det(IdV −f t) k≥0

and X

trace(Λk f )tk = det(IdV +f t)

k≥0

Proof. By passing to an algebraic closure of k, we may assume that V has a k-basis b1 , b2 , . . . , bd so that the matrix of f has upper triangular form

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3 Groups and Group Algebras

      

∗

λ1 λ2 ..

0

.

      

λd

where λi ∈ k are the eigenvalues of f . By (1.10) a basis of Symk V is given by the standard monomials of degree k, bI = bi1 bi2 . . . bik with I = (i1 , i2 , . . . , ik ) and 1 ≤ i1 ≤ i2 ≤ · · · ≤ ik ≤ d. We order the basis {bI }I by using the lexicographic ordering on the sequences I. Then
Symk f (bI ) = f (bi1 )f (bi2 ) . . . f (bik ) k   Y = λij bij + contributions from basis vectors bl with l < ij j=1

= λI bI + contributions from basis vectors bJ with J < I




where we have put λI = λi1 λi2 . . . λik . Thus, the matrix of Symk f for the basis {bI }I is upper triangular with the values λI on the diagonal. Consequently, X trace(Symk f ) = λI = hk (λ1 , λ2 , . . . , λd ) (3.33) I th

Here hk is the k complete symmetric polynomial, X def hk = hk (x1 , x2 , . . . , xd ) =

xi1 xi2 . . . xik

1≤i1 ≤i2 ≤···≤ik ≤d

X

=

l

l

l

x11 x22 . . . xdd

l1 +l2 +···+ld =k li ≥0

It is easy to see that the generating function of the polynomials hk is given by X

d Y

hk (x1 , s2 , . . . , xd )tk =

(1 + xi t + x2i t2 + . . . )

i=1

k≥0

d Y

1 = 1 − xi t i=1 From(3.33) and (3.34), we obtain X X trace(Symk f )tk = hk (λ1 , λ2 , . . . , λd )tk k≥0

k≥0

=

d Y

1 1 = 1 − λi t det(IdV −f t) i=1

(3.34)

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139

This proves the first formula. The proof of the second formula is similar. A basis of Λk V is given by the elements ∧bI = bi1 ∧ bi2 ∧ · · · ∧ bik with I = (i1 , i2 , . . . , ik ) and 1 ≤ i1 < i2 < · · · < ik ≤ d by (1.13). Using the lexicographic ordering on the sequences I, we again have
Λk f (∧bI ) = λI (∧bI ) + contributions from basis vectors ∧bJ with J < I Therefore, trace(Λk f ) = ek (λ1 , λ2 , . . . , λd )

(3.35)

th

where ek is the k elementary symmetric polynomial, X X Y def ek = ek (x1 , x2 , . . . , xd ) = xi1 xi2 . . . xik = xi 1≤i1 0 .

3.5 Some Applications to Invariant Theory We momentarily interrupt our development of group representation theory for a brief excursion into invariant theory, presenting some standard facts that are based on material from the previous sections. Let Γ be any group, not necessarily finite, and let V be a representation of kΓ , not necessarily finite-dimensional. Then Γ acts by graded k-algebra automorphisms on the tensor algebra TV , the symmetric algebra Sym V , and the exterior algebra ΛV . Thus, the homogeneous components V ⊗k , Symk V and Λk V all are kΓ representations (§3.3.4). Classical invariant theory is mostly concerned with the symmetric algebra and seeks to determine the structure of the subalgebra of Γ -invariants, M (Sym V )Γ = (Symk V )Γ k≥0

In practice, one often considers the dual representation V ∗ instead of V . From the viewpoint of representation theory, the difference is immaterial. As explained in Appendix C, Section C.3, the algebra O(V ) = Sym V ∗ for a finite-dimensional V is often referred to as the algebra of polynomial functions on V . The invariant subalgebra O(V )Γ is called an algebra of polynomial invariants. 3.5.1 Symmetric and Antisymmetric Tensors Recall from §3.3.4 that we have canonical epimorphisms of representations V ⊗n  Symn V ,

v1 ⊗ v2 ⊗ · · · ⊗ vn 7→ v1 v2 · · · vn

V ⊗n  Λn V ,

v1 ⊗ v2 ⊗ · · · ⊗ vn 7→ v1 ∧ v2 ∧ · · · ∧ vn

and

The symmetric group Sn also acts on V ⊗n , namely by place permutations: s.(v1 ⊗ v2 ⊗ · · · ⊗ vn ) = vs−1 1 ⊗ vs−1 2 ⊗ · · · ⊗ vs−1 n

(3.52)

The Sn -action manifestly commutes with the “diagonal” action of Γ on V ⊗n and it makes V ⊗n a kSn -representation. Assuming that char k - n!, we may apply the material of Section 3.4 with G = Sn . In particular, we may write M V ⊗n = (V ⊗n )(S) (3.53) S∈Irr kSn

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3 Groups and Group Algebras

and consider the projections (3.50). These projections commute with the action of Γ on V ⊗n and hence they are epimorphisms in Rep kΓ . We will describe the components (V ⊗n )(S) in (3.53) once we have learned more about Irr kSn ; see (4.51). For now, let us focus on the two degree-1 representations, 1 and sgn, of Sn and consider the projections (3.51) onto the corresponding weight spaces,

x

P

s∈Sn

(V ⊗n )(sgn)

∈

and

∈ 1 n!

A : V ⊗n

∈

(V ⊗n )(1)

∈

S : V ⊗n

x

s.x

1 n!

P

s∈Sn

sgn(s)s.x

The spaces (V ⊗n )(1) = (V ⊗n )Sn and (V ⊗n )(sgn) are commonly referred to as the spaces of symmetric and antisymmetric n-tensors, respectively, and the projection maps S and A as well as the corresponding primitive central idempotents of kSn are called symmetrizer and antisymmetrizer. Lemma 3.23. Let V ∈ Rep kΓ , where Γ is an arbitrary group, and assume that char k = 0 or char k > n. Then the canonical epimorphism V ⊗n  Symn V factors through the symmetrization S and its restriction to the space of symmetric n-tensors is an isomorphism in Rep kΓ , ∼ Symn V (V ⊗n )Sn −→

Similarly, the canonical epimorphism V ⊗n  Λn V factors through A and its restriction to the space of antisymmetric n-tensors is an isomorphism ∼ (V ⊗n )(sgn) −→ Λn V ⊗n Proof. The canonical epimorphism  Symn V has kernel I ∩ V ⊗n , where
V 0 0 0 I = v ⊗ v − v ⊗ v | v, v ∈ V is as in §1.1.2. Since V ⊗n = (V ⊗n )(1) ⊕ Ker S , it suffices to show that I ∩ V ⊗n = Ker S . If x ∈ Ker S , then x = x − S x = P 1 s∈Sn (x − s.x). Since Sym V is commutative, x and each s.x have the same n! image in Symn V . It follows that x maps to 0 ∈ Symn V , whence I ∩ V ⊗n ⊇ Ker S . For the reverse inclusion, it suffices to show that S x = 0 for every element of the form x = y ⊗ (v ⊗ v 0 − wv ⊗ v) ⊗ z with y ∈ V ⊗r and z ∈ V ⊗(n−r−2) . Note that the transposition t = (r + 1, r + 2) ∈ Sn satisfies t.x = −x. Since Sn = An t An t, we obtain 1 X 1 X (s.x + st.x) = (s.x − s.x) = 0 Sx =

n!

s∈An

n!

s∈An

as desired.
The argument for Λn V is analogous, using the ideal J = v ⊗ v | v ∈ V of TV in place of I and the antisymmetrizer A instead of S . The inclusion Ker A ⊆ J ∩ V ⊗n follows from the anticommutativity property (1.12) of the exterior algebra

3.5 Some Applications to Invariant Theory

155

ΛV : the images of x and sgn(s)s.x in ΛV are identical for all x ∈ V ⊗n , s ∈ Sn . For J ∩ V ⊗n ⊆ Ker A , it suffices to show that A x = 0 for elements of the form x = y ⊗ v ⊗ v ⊗ z with y ∈ V ⊗r and z ∈ V ⊗(n−r−2) . Now the transposition t = (r + 1, r + 2) ∈ Sn satisfies t.x = x and so sgn(s)s.x + sgn(st)st.x = sgn(s)s.x − sgn(s)s.x = 0 for all s. Therefore, 1 X (sgn(s)s.x + sgn(st)st.x) = 0 Ax= n!

s∈An

t u

which completes the proof.

The reader may wish to have a look at Exercise 3.15 which interpretes the iso∼ Symn V as an isomorphism between Sn -invariants and Sn morphism (V ⊗n )Sn −→ coinvariants. The special case n = 2 of Lemma 3.23 will be important several times later on. So let us note for future reference that, for char k 6= 2, the decomposition (3.53) can be written as 2 2 V ⊗2 = (V ⊗2 )(1) ⊕ (V ⊗2 )(sgn) ∼ = Sym V ⊕ Λ V

(3.54)

The Finite-Dimensional Case We will now focus on the case where V is finite-dimensional. Observe that the nth tensor powers v ⊗n = v ⊗ v ⊗ · · · ⊗ v ∈ V ⊗n are evidently symmetric. Our goal is to show that, if the base field k is large enough, then the space (V ⊗n )Sn of symmetric n-tensors is generated by nth tensor powers. The proof will make use of some basic facts concerning polynomial functions and the Zariski topology on V ; see Appendix C, Section C.3. Proposition 3.24. Let V be a finite-dimensional k-vector space.

 (a) If |k| ≥ n, then (V ⊗n )Sn = v ⊗n | v ∈ V k ; in fact, it suffices to let v range over any given Zariski dense subset of V .

 (b) Let 0 6= f ∈ Om (V ). If |k| ≥ n + m, then (V ⊗n )Sn = v ⊗n | v ∈ Vf k . Proof. We begin with some preliminary observation concerning the structure of V ⊗n as Sn -representation. Fix a k-basis {xi }d1 of V . Then the monomials xI = xi1 ⊗ xi2 ⊗ · · · ⊗ xin with I = (i1 , . . . , in ) ∈ X := [d]n form a k-basis of V ⊗n that is permuted by the Sn -action: s.xI = xs.I with s.I = (is−1 1 , . . . , is−1 n ). Therefore, V ⊗n ∼ = kX is a permutation representation of Sn . A transversal for the orbit set Sn \X is given by the sequences Im = (1, . . . , 1, 2, . . . , 2, . . . , d, . . . , d) | {z } | {z } | {z } m1

m2

md

with m = (m1 , m2 , . . . , md ) ∈ Zd+ and |m| := mj = n. The isotropy group of Im is the subgroup of Sn consisting of those permutation of [n] that stabilize all P

156

3 Groups and Group Algebras

subsets {1, . . . , m1 }, {m1 + 1, . . . , m1 + m2 }, . . . , {m1 + · · · + md−1 + 1, . . . , n}. Thus, denoting this subgroup by Sm , we have an isomorphism M V ⊗n ∼ k[Sn /Sm ] (3.55) = d

m∈Z+ : |m|=n

Now (3.23) provides us with a k-basis of (V ⊗n )Sn , namely the orbit sums X σm = xs.Im s∈Sn /Sm

In both (a) and (b), our goal is to show that the vector space (V ⊗n )Sn is generated by the elements v ⊗n with v ranging over a certain given subset D ⊆
V . This in ∗ turn amounts to showing that, for any linear form 0 6= l ∈ (V ⊗n )Sn , there is Pd some v ∈ D such that hl, v ⊗n i = 6 0. So let us fix l and write v = i=1 λi xi . The development of v ⊗n in terms of the above basis {σ m } is as follows: X m m m m v ⊗n = λ σ m with λm = λ1 1 λ2 2 · · · λd d d

m∈Z+ |m|=n

P Put h = h(l) := |m|=n xm hl, σ m i ∈ On (V ), where xm = (x1 )m1 · · · (xd )md , and note that h 6= 0, because hl, σ m i 6= 0 for some m and the standard monomials xm are k-independent. Moreover, writing Φ : O(V ) → kV = {functions V → k} as in (C.1), we have Φ(h)(v) = hl, v ⊗n i: V

k

v

∈

∈

Φ(h) :

P

|m|=n

λm hl, σ m i = hl, v ⊗n i

Thus, our goal is to show that Φ(h)(v) 6= 0 for some v ∈ D or, equivalently, D ∩ Vh 6= ∅. (a) Our hypothesis |k| ≥ n ensures that Φ(h) 6= 0 (Exercise C.2). Therefore, h(v) 6= 0 for some v ∈ V and so Vh is a nonempty open subset of V . Thus, D∩Vh 6= ∅ for D = V , which proves the first assertion of (a). Moreover, for any Zariski dense subset D ⊆ V , we have D ∩ Vh 6= ∅, proving the second assertion. (b) Now our goal now is to show that Vf ∩ Vh 6= ∅. But 0 6= f h ∈ Om+n (V ) and so Φ(f h) 6= 0 by our hypothesis on k (Exercise C.2 again). Therefore, ∅ 6= Vf h = Vf ∩ Vh as desired. This completes the proof of the proposition. t u The permutation representations k[Sn /Sm ] as in the proof above are often called Young modules and the groups Sm are called Young subgroups of Sn . 2 More generally, any partition X = X1 t X2 t · · · t Xl of a set X gives rise to a Young 2

Both are named after Alfred Young (1873–1940).

3.5 Some Applications to Invariant Theory

157

subgroup Y ≤ SX , consisting of those permutations of X that stabilize all Xi ; this group is isomorphic to the direct product of the permutation groups of the parts of the partition, Y ∼ = SX1 × SX2 × · · · × SXl We may surely remove all Xi = ∅ from the partition without changing Y . In particular, Sm only depends on the sequence of positive integers, with sum equal to n, that is obtained from the d-tuple m = (m1 , m2 , . . . , md ) ∈ Zd+ by omitting all mi = 0. Such sequences are called compositions of n. 3.5.2 Polarization Turning to the dual representation V ∗ , the homogeneous component Symn V ∗ is called the space of homogeneous forms of degree n on V . The space (V ⊗n )∗ is canonically isomorphic to the space MultLin(V n , k) of all multilinear maps V n = V ×V ×· · ·×V −→ k by (B.12). We continue to assume that V is finite-dimensional. ⊗n ∗ ∗ ⊗n ∼ n Then (V ∗ )⊗n ∼ = (V ) (Exercise 3.20) and hence (V ) = MultLin(V , k). Letting h · , · i denote evaluation, the isomorphism is explicitly given by ∼

MultLin(V n , k) (3.56)

∈

∈

(V ∗ )⊗n



f1 ⊗ f2 ⊗ · · · ⊗ fn

 Q (v1 , v2 , . . . , vn ) 7→ i hfi , vi i

Q Writing hf, vi = i hfi , vi i for f = f1 ⊗ f2 ⊗ · · · ⊗ fn and v = (v1 , v2 , . . . , vn ), the Sn -action (3.52) on (V ∗ )⊗n and the corresponding action on V n , s.v = (vs−1 1 , vs−1 2 , . . . , vs−1 n ) evidently satisfy hf, vi = hs.f, s.vi for all s ∈ Sn . Thus, under the isomorphism (3.56), the Sn -invariants ((V ∗ )⊗n )Sn correspond to the symmetric multilinear maps, that is, multilinear maps V n → k that are constant on Sn -orbits in V n . If char k = 0 or char k > n, then we obtain the isomorphism Symn V ∗

∼ Lemma 3.23

(V ∗ )⊗n


Sn

∼ (3.56)

n

symmetric multilinear n maps V −→ k

o

(3.57)

In the invariant theory literature, this isomorphism is called polarization; see Weyl [182, p. 5ff] or Procesi [147, p. 40ff]. Below, we carry this process out explicitly for the elementary symmetric functions. For more on polarization and and its inverse, called restitution, see Exercise 3.39. Example 3.25 (Polarization of the elementary symmetric functions). Fixing a basis x1 , . . . , xd of V ∗ , we may consider the nth elementary symmetric polynomial for n ≤ d,

158

3 Groups and Group Algebras

xi1 xi2 · · · xin ∈ Symn V ∗

X

en = en (x1 , x2 , . . . , xd ) =

1≤i1 n} and so . . . ⊆ Vn−2 ⊆ Vn−1 ⊆ . . . . Thus, Vn−2 provides us with an irreducible component of Vn−1↓Sn−1 . The vector vn−1 =

n−1 X

n−1 X

i=1

i=1

(bi − bn ) =

bi − (n − 1)bn ∈ Vn−1

is a nonzero Sn−1 -invariant that does not belong to Vn−2 . For dimension reasons, we conclude that Vn−1↓Sn−1 = kvn−1 ⊕ Vn−2 ∼ = 1Sn−1 ⊕ Vn−2 . We deduce inductively Ln−1 that Vn−1 = j=1 kvj and that {v1 , . . . , vn−1 } is the GZ-basis of Vn−1 . Of course, any rescaling of this basis would be another GZ-basis; we shall discuss other choices later (Examples 4.17 and 4.19). It is straightforward to check that the action of the Coxeter generator si = (i, i + 1) ∈ Sn on the GZ-basis {vj } is as follows:   for j 6= i − 1, i vj (4.8) si .vj = 1i vi−1 + (1 − 1i ) vi for j = i − 1   1 1 (1 + i )vi−1 − i vi for j = i These equations determine the GZ-basis {vj } up to a common scalar factor: if (4.8) also holds with wj in place of vj , then vj 7→ wj is an Sn -equivariant endomorphism of Vn−1 and hence an element of D(Vn−1 ) = k. 4.2.4 Maximality of GZ n We now use the foregoing to derive further information about Gelfand-Zetlin algebra GZ n . Theorem 4.4. GZ n consists of all a ∈ kSn such that the GZ-basis of each V ∈ Irr Sn consists of eigenvectors for aV . Consequently, GZ n is a maximal com×d mutative subalgebra of kSn and GZ n is semisimple: GZ n ∼ = k n with dn = P V ∈Irr Sn dim V . Proof. Fixing the GZ-basis {vT } for each V ∈ Irr kSn , we identify Endk (V ) with Matdim V (k). The isomorphism (4.4) then identifies kSn with the direct product of these matrix algebras. Let D denote the subalgebra of kSn that corresponds to the direct product of the algebras of diagonal matrices in each component. Our main goal is to show that D = GZ n : Y ∼ kSn Matdim V (k) V ∈Irr Sn

GZ n

(4.9)

⊆

⊆

via GZ-bases

∼

!

k

Y V ∈Irr Sn

..

. k

4.2 The Branching Graph

189

×d The isomorphism GZ n ∼ = k n is then clear, and so is the maximality assertion. Indeed, the subalgebra of diagonal matrices in any matrix algebra is self-centralizing: the only matrices that commute with all diagonal matrices are themselves diagonal. Therefore, D is a self-centralizing subalgebra of kSn , and hence it is a maximal commutative subalgebra. In particular, in order to prove the equality D = GZ n , it suffices to show that D ⊆ GZ n , because we already know that GZ n is commutative. It remains to show that D ⊆ GZ n . Let e(V ) ∈ Zn = Z (kSn ) denote the primitive central idempotent corresponding to a given V ∈ Irr Sn (§1.4.3). Recall that, for any M ∈ Rep Sn , the operator e(V )M projects M onto the V -homogeneous component M (V ) and annihilates all homogeneous components of M other than M (V ). Therefore, if T : 1S1 = W1 → W2 → · · · → Wn = V is a path in B, then the element e(T ) := e(W1 )e(W2 ) · · · e(Wn ) ∈ k[Z1 , Z2 , . . . , Zn ] = GZ n acts as the projection πT : V  VT = kvT in (4.6), and e(T ) acts as 0V 0 on all V 6= V 0 ∈ Irr Sn . Thus, in (4.9), we have

kSn

∼

Y

Matdim V (k)

∈

∈

V ∈Irr Sn

e(T )

(0, . . . , πT , . . . , 0)

This shows that the idempotents e(T ) form the standard basis of the diagonal algebra D, which proves the desired inclusion D ⊆ GZ n . t u The fact that the GZ-basis {vT } of any V ∈ Irr Sn consists of eigenvectors for GZ n means explicitly that a.vT = φT (a)vT

(a ∈ GZ n )

for suitable φT ∈ HomAlgk (GZ n , k). Using the terminology of Example 1.28, this equation states that vT is a weight vector of weight φT for the action of the algebra GZ n . Moreover, then number dn in Theorem 4.4 is equal to the total number of paths in B from 1S1 to a level-n vertex ∈ Irr Sn (Exercise 4.5) and the isomorphism ∼ GZ n −→ k×dn is given by a 7→ (φT (a))T . Thus, n T a path 1 → . . . in B o S1 HomAlgk (GZ n , k) = φT (4.10) with endpoint ∈ Irr Sn

Exercises 4.4 (Bottom of the branching graph B). Verify the bottom of the branching graph B as in Figure 4.1. The notation is as in §3.6.2: W and W ± are the 5-dimensional irreducible representations of S5 , and Ve2 is the standard representation of S3 inflated along the epimorphism S4  S3 .

190

4 The Symmetric Groups

4.5 (Dimension of the Gelfand-Zetlin algebra). Let dn = dim GZ n as in Theorem 4.4. (a) Show that dn is equal to the total number of paths in B from 1S1 to a level-n vertex ∈ Irr Sn and dn is also to the composition length of the regular representation (kSn )reg by (4.9). (b) Show that the first five values of the sequence dn are: 1, 2, 4, 10, 26.1 4.6 (Lengths of homogeneous components). Let V ∈ Irr Sn and W ∈ Irr Sk be given and assume that k ≤ n. Show that the length of the W -homogeneous component of V ↓Sk is equal to the number of paths W → · · · → V in B. 4.7 (Orthogonality of GZ-bases). Let V ∈ Irr Sn and let ( · , · ) : V × V → k be any bilinear form which is Sn -invariant, that is, (s.v, s.v 0 ) = (v, v 0 ) for all v, v 0 ∈ V and s ∈ Sn . Show that every GZ-basis {vT } of V must be orthogonal: (vT , vT 0 ) = 0 for T 6= T 0 . [Use the fact that representations of symmetric groups are self-dual by Lemma 3.27.] 4.8 (Weights of the standard representation). Let Vn−1 be the standard representation of Sn and let vj = b1 + b2 + · · · + bj − jbj+1 as in Example 4.3. Show that vj has weight (0, 1, . . . , j − 1, −1, j, . . . , n − 2).

4.3 The Young Graph We now start afresh, working in combinatorial rather than representation theoretic territory. 4.3.1 Partitions and Young Diagrams Consider the familiar set of partitions of n: n o X Pn = λ = (λ1 , λ2 , . . . ) | λi ∈ Z+ , λ1 ≥ λ2 ≥ λ3 . . . and λi = n i

P We will usually write λ ` n in place of λ ∈ Pn and we also put |λ| = i λi . Partitions will be visualized by Young diagrams: the Young diagram of λ = (λ1 , λ2 , . . . ) consists of rows of boxes that are aligned on the left, with λ1 boxes in the first row, λ2 in the second, etc. The unique partition with |λ| = 0 thus has an empty Young diagram. We will generally only consider partitions with |λ| ≥ 1 (but see Exercise 4.9). Here, for example, is the Young diagram of the partition (7, 5, 4, 4, 2) ` 22:

Young diagrams are also called Ferrers diagrams, particularly when represented using dots instead of boxes. 1

For more information on this sequence, see the On-Line Encyclopedia of Integer Sequences [165, Sequence A000085].

4.3 The Young Graph y

x

191

The columns and rows of a given Young diagram will be numbered left-to-right and top-to-bottom, respectively, starting with number 1. The box in the xth row and y th column will also be referred to as the box in position (x, y) or the (x, y)-box.

Reflecting a given partition λ ` n across the line y = x yields the so-called conjugate partition; it will be denoted by λc . For example, (7, 5, 4, 4, 2)c = (5, 5, 4, 4, 2, 1, 1).

λ

c

λ

4.3.2 The graph Y and the Graph Isomorphism Theorem The Young graph Y has vertex set G vert Y = Pn n≥1 removable box

Each λ ∈ vert Y will be represented by its Young diagram. An arrow µ → λ in Y will mean that the Young diagram of µ is obtained from the one of λ by removing just one box, necessarily a “southeast" corner box. Note that the number of removable boxes of λ = (λi )i≥1 is equal to the number of distinct values among the λi . Formally, writing µ ≤ λ for λ = (λi ), µ = (µi ) ∈ vert Y if µi ≤ λi holds for all i , an arrow µ → λ means that µ < λ but there is no ν ∈ vert Y with µ < ν < λ.

P5 : P4 : P3 : P2 : P1 : Fig. 4.2. Bottom of the Young graph Y

We will refer to Pn as the set of level-n vertices of Y. Figure 4.2 shows the first five levels of the Young graph. Comparison with Figure 4.1 shows a striking similarity to the first five levels of the branching graph B. In fact, we have the following fundamental result.

192

4 The Symmetric Groups

Graph Isomorphism Theorem. The graphs Y and B are isomorphic. Explicitly, the Graph Isomorphism Theorem states that there is a bijection ∼ φ : vert Y −→ vert B such that there is an arrow µ → λ in Y if and only if there ∼ is an arrow φ(µ) → φ(λ) in B. We will then write φ : Y −→ B. We may of course ∼ also speak of automorphisms B −→ B and similarly for the graph Y. In fact, it is not hard to see that conjugation, λ 7→ λc , is the only non-identity automorphism of Y ∼ (Exercise 4.10). Thus, there are at most two possible isomorphisms Y −→ B. The proof of the Graph Isomorphism Theorem will be given in Section 4.4. Here, we just point out a few immediate consequences of the existence of a graph isomor∼ phism Y −→ B. Let us write the bijection on vertices as λ 7−→ V λ For example, we clearly must have V = 1S1 , because and 1S1 are the sole ver∼ tices of Y and B with no incoming arrows. More generally, the isomorphism Y −→ B will bijectively map the n-vertex paths → µ2 → · · · → µn in Y to the corresponding paths in B, and hence it will give match the level-n vertices of Y with those of ∼ B, giving bijections Pn −→ Irr Sn for all n. Of course, we already know that such bijections exists: there are as many irreducible representations of Sn as there are conjugacy classes of Sn (Corollary 3.20) and the conjugacy classes in turn are in bijection with the partitions of n (§3.6.2). The full form of the the Graph Isomorphism Theorem will require more work, however. 4.3.3 Some Consequences of the Graph Isomorphism Theorem The Graph Isomorphism Theorem will allow us to study the irreducible representations of all Sn in terms of simpler combinatorial information coming from Y. Here are some examples. The branching rule Since µ → λ in Y is equivalent to V µ → V λ in B, we may rewrite formula (4.5) as V λ↓Sn−1 ∼ =

M

Vµ

(4.11)

µ→λ in Y

This formula is referred to as a branching rule. The number of arrows µ → λ in (4.11) is equal to the number of removable boxes of λ, which in turn is equal to the number of distinct values (row lengths in the Young diagram) of λ. Therefore, length V λ↓Sn−1 = #{distinct values of λ} In particular, V λ ↓Sn−1 is irreducible if and only if the Young diagram of λ is a rectangle. The branching rule (4.11) has the following reformulation:

4.3 The Young Graph

V µ↑Sn ∼ =

M

Vλ

193

(4.12)

µ→λ in Y

Indeed, (4.11) says that the multiplicity of V µ in V λ↓Sn−1 is equal to 1 if there is an arrow µ → λ in Y and equal to 0 otherwise; (4.12) makes the same statement about the multiplicity of V λ in V µ↑Sn . The equivalence of (4.11) and (4.12) thus follows from Frobenius reciprocity (Corollary 1.37): m(V µ , V λ↓Sn−1 ) = m(V λ , V µ↑Sn ). Dimension ∼ Any graph isomorphism Y −→ B will induce bijections between the set of all paths → · · · → λ in Y and the set of all paths 1S1 → · · · → V λ in B. Since the size of the latter set of paths equals dim V λ by (4.7), we obtain

n def dim V λ = f λ = # paths

→ · · · → λ in Y

o

(4.13)

The number f λ will be determined in §4.3.5 below. ThePdimension dn = dim GZ n (Theorem 4.4 and Exercise 4.5) can be written as dn = λ`n f λ . 4.3.4 Paths in Y and Standard Young Tableaux In this subsection, we give another description of the number f λ defined in (4.13) in terms of standard Young tableaux. By definition, a standard Young tableau of shape λ ` n, or λ-tableau for short, is obtained by filling the numbers 1, 2, . . . , n into the boxes of the Young diagram of λ in such a way that the numbers increase along rows (left to right) and along columns (top to bottom). Clearly, the (1, 1)-box must contain the number 1 and n must occur in some removable corner box. Removing this box, we obtain the Young diagram of a partition µ with µ → λ. Continuing in this manner, successively removing the boxes containing the highest number, we eventually end up with the tableau 1 . This process is easily seen to yield a bijection o n o n ∼ λ-tableaux (4.14) paths → · · · → λ in Y ←→ Just as we have identified partitions with their Young diagrams in the description of the Young graph Y, we will also permit ourselves to view the above bijection as an identification. Thus, we will not distinguish between paths in Y and standard Young tableaux. In particular, we may rewrite the definition of f λ in (4.13) as follows: n o f λ = # λ-tableaux

(4.15)

194

4 The Symmetric Groups

Example 4.5. Here are all standard Young tableaux of shape λ = (2, 2, 1) ` 5 along with the corresponding paths in Y: 1 2 3 4 5

:

−→

−→

−→

−→

1 3 2 4 5

:

−→

−→

−→

−→

1 2 3 5 4

:

−→

−→

−→

−→

1 3 2 5 4

:

−→

−→

−→

−→

1 4 2 5 3

:

−→

−→

−→

−→

4.3.5 The Hook-Length Formula Identifying a given λ ` n with its Young diagram as usual, we consider the hook at a given box x of λ as in the picture on the right and we define the hook length by

x . . .

...

def

h(x) = #{boxes in the hook at x} Note that corner boxes or, equivalently, removable boxes are exactly those boxes, x, with h(x) = 1. The following formula is due to Frame, Robinson and Thrall [61]. n! x∈λ h(x)

Hook-Length Formula. For λ ` n, we have f λ = Q

For example, consider the partition λ = (3, 2, 2, 1) ` 8. Filling each box in the Young diagram of λ with the length of the hook at this box, we obtain the scheme 6 4 1 4 2 3 1 1

Hence, the hook formula tells us that f λ =

8! 2 3 64 321

= 70.

We shall present a probabilistic proof of the hook-length formula below that is due to Greene, Nijenhuis and Wilf [76]. Here is some brief background on probability and the description of the particular experiment that is used in the proof. The hook-walk experiment. Suppose that a certain experiment has a finite set Ω of possible outcomes; the set Ω is usually called the “sample space” and subsets E ⊆ Ω are referred to as “events”. Assume further that, for each ω ∈ Ω, there is a probability P value p(ω) ∈ R≥0 such that ω∈Ω p(ω) = 1 . Then the probability of a given event E ⊆ Ω is defined by X def P (E) = p(ω) . ω∈E

4.3 The Young Graph

195

|E| For example, if all outcomes ω ∈ Ω have the same probability, then P (E) = |Ω| . The particular experiment that we will consider below is an example of a memoryless random walk or Markov chain: each step in the walk depends only on the current position and not on the sequence of steps that preceded it. Specifically, consider a partition λ ` n, identified with its Young diagram. To start with, choose a box, x = x0 , among the n boxes of λ with uniform probability n1 . If x is a corner box, then stop; otherwise, choose a different box, x1 6= x, in the hook at x with uni1 form probability q(x) := h(x)−1 . If x1 is a corner box, then stop; otherwise, choose a different box, x2 6= x1 , in the hook at x1 with uniform probability q(x1 ) etc. Each step moves either down or right, and the walk will terminate at some corner box xt = c ∈ λ after finitely many steps:

ω : x = x0 → x1 → · · · → xt−1 → xt = c We will refer to ω as a “hook walk” in λ . Our sample set Ω consists of all such hook walks; this is clearly a finite set. The probability of the hook walk ω is given by p(ω) =

1 q(ω) n

with q(ω) := q(x0 )q(x1 ) · · · q(xt−1 )

(4.16)

P It is not hard to see that ω∈Ω p(ω) = 1. For each corner box c, consider the event Ec = {all hook walks in λ that end at c}. These events form a partition of the sample set Ω. Therefore, X X X1 X q(ω) (4.17) 1= p(ω) = P (Ec ) = ω∈Ω

(4.16)

c

c

n

ω∈Ec

where c runs over the corner boxes of λ. Proof of the hook-length formula. Our goal is to prove the formula f λ = where we have put Y H(λ) := h(x)

n! H(λ) ,

x∈λ

We proceed by induction on n . The formula is trivially true for n = 1. To deal with n > 1, we use the recursion X µ fλ = f µ→λ λ

which is evident from the definition of f as the number of paths → · · · → λ in Y. By induction, we know that f µ = (n−1)! H(µ) for all µ in the above sum. Thus, we need P P (n−1)! H(λ) n! to show that H(λ) = µ→λ H(µ) or, equivalently, 1 = µ→λ n1 H(µ) . Recall that µ → λ means that µ arises from λ by removing one corner box, say c. Denoting the resulting µ by λ \ c, our goal is to show that 1=

H(λ) n H(λ \ c)

X1 c

196

4 The Symmetric Groups

where c runs over the corner boxes of λ. Comparison with (4.17) shows that it suffices to prove the equality X H(λ) q(ω) = (4.18) H(λ \ c)

ω∈Ec

for each corner box c. First, let us consider the right-hand side of (4.18). Note that λ \ c has the same hooks as λ, except that the hook at c, of length 1, is missing and the hooks at all boxes in the gray region marked B = B(c) on the right have lengths shorter by 1 than the corresponding hooks of λ. Therefore, the right hand side of (4.18) can be written as follows:

B

B

c

Y h(b) H(λ) = H(λ \ c) h(b) − 1 b∈B

1 from the hook-walk experiment, we can write the Using the notation q(b) = h(b)−1 Q P Q product on the right as b∈B (1 + q(b)) = S⊆B b∈S q(b). Hence,

X Y H(λ) = q(b) H(λ \ c)

(4.19)

S⊆B b∈S

B x

B

x

0

00

c

x

Now for the left-hand side of (4.18). For each hook walk ω ∈ Ec , let Sω ⊆ B denote the set of boxes that arise as the horizontal and vertical projections of the boxes of ω into B; see the green boxes on the left. Note that, while Sω generally does not determine the entire walk ω, the starting point, x, is certainly determined, and if x ∈ B, then ω is determined. We claim that, for each subset S ⊆ B, X Y q(ω) = q(b) (4.20) ω∈Ec Sω =S

b∈S

This will give the following expression for the left-hand side of (4.18): X X X X Y q(ω) = q(ω) = q(b) ω∈Ec

S⊆B ω∈Ec Sω =S

S⊆B b∈S

By (4.19) this is identical to the right-hand side of (4.18), proving (4.18). It remains to justify the claimed equality (4.20). For this, we argue by induction on |S|. The only hook walk ω ∈ Ec with Sω = ∅ is the “lazy walk” starting and

4.3 The Young Graph

197

ending at c; so the claim is trivially true for |S| = 0. The claim is also clear if the starting point, x, of ω belongs to B, because then the sum on the left has only one term, which is equal to the right-hand side of (4.20). So assume that x ∈ / B ∪ {c}. Then there are two kinds of possible hook walks ω ∈ Ec with Sω = S: those that start with a move to the right and those that start with a move down. Letting η denote the remainder of the walk and letting x0 , x00 ∈ B be the vertical and horizontal projections of x into B, we have Sη = S \{x0 } in the former case and Sη = S \{x00 } in the latter. Therefore, by induction, ! X X X q(ω) = q(x) q(η) + q(η) ω∈Ec Sω =S

η∈Ec 0 Sη =S\{x }

η∈Ec 00 Sη =S\{x }

! Y

= q(x)

q(b)

+

0

Y

q(b) 00

b∈S\{x }

b∈S\{x }

0 00
Q )+h(x )−2 Q = h(x h(x)−1 The last expression can be written as q(x) q(x1 0 ) + q(x100 ) Q Q with = b∈S q(b), the right-hand side of (4.20). Thus, we wish to show that 0

00

= 1 or, equivalently, h(x) + 1 = h(x0 ) + h(x00 ). But this is indeed true, as the following picture shows: h(x )+h(x )−2 h(x)−1

x

x

00

0

x

This finishes the proof of the hook-length formula.

t u

Exercises L 4.9 (Up and down operators). Let Z[Y] = n Z[Pn ] denote the set of all formal Z-linear combinations of partitions λ. Here, Z[Pn ] = 0 for n < 0, because Pn is empty in this case, and Z[P0 ] ∼ = Z, because P0 = {∅}. Thus, we have added the empty Young diagram ∅ as a root to Y, with a unique arrow P ∅ → . Consider the operators U, D ∈ End (Z[Y]) that are defined by U (λ) = Z λ→µ µ and D(λ) = P µ. Show that these operators satisfy the Weyl algebra relation DU = U D+1. µ→λ 4.10 (Automorphisms of Y). (a) Show that each λ ∈ Pn (n > 2) is determined by the set S(λ) := {µ ∈ Pn−1 | µ → λ in Y}. (b) Conclude by induction on n that the graph Y has only two automorphisms: the identity and conjugation.

198

4 The Symmetric Groups

4.11 (Rectangle partitions). Show:
th 2n 1 (a) f (n,n) = n+1 n , the n Catalan number. (b) f λ > rc for λ = (cr ) := (c, . . . , c) with c, r ≥ 2 and rc ≥ 8. | {z } r

4.12 (Dimensions of the irreducible representations of S6 ). Extend the bottom of Y from Figure 4.2 up to layer P6 . Find the degrees of all irreducible representations of S6 . 4.13 (Hook partitions and exterior powers of the standard representation). Let ∼ λ 7→ V λ be the bijection on vertices that is given by a graph isomorphism Y −→ B 7→ 1S2 . Show as in §4.3.2. Assume that the isomorphism has been chosen so that k

that V (n−k,1 ) = Λk Vn−1 holds for all n and k = 0, . . . , n − 1, where (n − k, 1k ) = (n − k, 1, . . . , 1) are the “hook partitions". | {z } k

4.14 (Irreducible representations of dimension < n). Let n ≥ 7. Show that 1, sgn, ± the standard representation Vn−1 , and its signed twist Vn−1 are the only irreducible representations of degree < n of Sn . [Use the existence of a graph isomorphism Y∼ = B and Exercise 4.11(b).]

4.4 Proof of the Graph Isomorphism Theorem The main goal of this section is to provide the proof of the Graph Isomorphism Theorem; it will be given in Corollary 4.13 after some technical tools have been deployed earlier in this section. In short, the strategy is to set up a bijection between the collection of all paths → . . . in Y with endpoint in Pn and the seemingly rather more complex collection of paths 1S1 → . . . in B with endpoint in Irr Sn . Having already identified paths in Y with standard Young tableaux in (4.14), we will show in §4.4.1 that standard Young tableaux with n boxes in turn are in oneto-one correspondence with a certain set of n-tuples Cont(n) ⊆ Zn . Similarly, for the paths in B, we will also construct a certain set of n-tuples, Spec(n), which is a specific realization of Spec GZ n using the JM-elements. Finally, we will show that Cont(n) = Spec(n), thereby obtaining a bijection n

paths → . . . in Y with endpoint ∈ Pn

o

∼

Cont(n) = Spec(n)

∼

n

paths 1S1 → . . . in B with endpoint ∈ Irr Sn

o

4.4.1 Content We define the content of a standard Young tableau T with n boxes to be the n-tuple cT = (cT,1 , cT,2 , . . . , cT,n ), where cT,i = a means that the box of T containing the number i lies on the line y = x + a; we will call this line the a-diagonal. Thus, cT,i = (column number) − (row number) where i occurs in T

4.4 Proof of the Graph Isomorphism Theorem

Since the number 1 must occupy the (1, 1)-box of any standard Young tableau, we always have cT,1 = 0. It is easy to see that any standard Young tableau is determined by its content. Indeed, we can reconstruct T from the fibers c−1 T (a)

:= {i | cT,i = a}

(a ∈ Z)

199 y

i

y =x+a x

The size of these fibers tell us the shape of T or, equivalently, the underlying Young −1 diagram: there must be |c−1 T (a)| boxes on the a-diagonal. The elements of cT (a) give the content of these boxes: we must fill them into the boxes on the a-diagonal in increasing order from top to bottom. Example 4.6. Suppose we are given cT = (0, 1, −1, 2, 3, −2, 0, −1, 1). The nonempty −1 −1 −1 fibers are c−1 T (−2) = {6}, cT (−1) = {3, 8}, cT (0) = {1, 7}, cT (1) = {2, 9}, −1 −1 cT (2) = {4} and cT (3) = {5}. Thus, the resulting standard Young tableau T is 4 3 2 1 −1 −2 −3

0

1 2 4 5 3 7 9 6 8

Of course, not every n-tuple of integers is the content of a standard Young tableau with n boxes. In fact, there are only finitely many such Young tableaux and we also know that the first component of any content vector must always be 0. Our next goal will be to give a description of the following set: def  Cont(n) = cT | T is a standard Young tableau with n boxes ⊆ Zn

Example 4.7. We list all standard Young tableaux with 4 boxes and their content vectors; these form the set Cont(4). Note that, quite generally, reflecting a given standard Young tableau T across the diagonal y = x results in another Young tableau, T c , satisfying cT c = −cT . 1 2 3 4 (0, 1, 2, 3)

1 2 3 4 (0, −1, −2, −3)

1 3 4 2 (0, −1, 1, 2)

1 2 3 4 (0, 1, −1, −2)

1 2 4 3 (0, 1, −1, 2)

1 3 2 4 (0, −1, 1, −2)

1 2 3 4 (0, 1, 2, −1)

1 4 2 3 (0, −1, −2, 1)

1 3 2 4 (0, −1, 1, 0)

1 2 3 4 (0, 1, −1, 0)

200

4 The Symmetric Groups

An equivalence relation As we have remarked above, the shape of a standard Young tableau T is determined by the multiplicities |c−1 T (a)| for a ∈ Z. Thus, two standard Young tableaux T and T 0 have the same shape if and only if cT 0 = scT for some s ∈ Sn , where Sn acts on n-tuples by place permutations: s(c1 , . . . , cn ) := (cs−1 1 , . . . , cs−1 n ) Equivalently, T and T 0 have the same shape if and only if T 0 = sT for some s ∈ Sn , where sT is obtained from T by replacing the entries in all boxes by their images under s. The reader will readily verify (Exercise 4.16) that csT = scT

(4.21)

Define an equivalence relation ∼ on Cont(n) by def

cT ∼ cT 0 ⇐⇒ cT 0 differs from cY only by a place permutation ⇐⇒ T and T 0 have the same shape ⇐⇒ T and T 0 describe paths

→ . . . in Y with the same endpoint

(4.14)

We summarize the correspondences discussed in the foregoing: n

Cont(n) ∼

standard Young tableaux with n boxes

forget places

Cont(n)/∼

∼

o

n

∼

remember endpoint

empty boxes

n

Young diagrams with n boxes

o

paths → . . . in Y with endpoint ∈ Pn

∼

o (4.22)

Pn

Admissible transpositions Not all place permutations of the content vector cT of a given standard Young tableau T necessarily produce another content vector of a standard Young tableau. For one, the first component must always be 0. The case of the Coxeter generators si = (i, i + 1) ∈ Sn will be of particular interest. The transposition si swaps the boxes of T containing i and i + 1 while leaving all other boxes untouched and si (c1 , . . . , cn ) = (c1 , . . . , ci−1 , ci+1 , ci , ci+2 , . . . , cn ). It is easy to see (Exercise 4.16) that si T is another standard Young tableau if and only if i and i + 1 occur in different rows and columns of T . In this case, si will be called an admissible transposition for T . Note that the boxes containing i and i + 1 belong to different rows and columns of T if and only if these boxes are not direct neighbors in T , and this is also equivalent to the condition that the boxes of T containing i and i + 1 do not lie on adjacent diagonals, that is, cT,i+1 6= cT,i ± 1

(4.23)

4.4 Proof of the Graph Isomorphism Theorem

201

...

For a given partition λ ` n, consider the particular standard Young tableau T (λ) that is obtained from the Young diagram of λ by filling the numbers 1, 2, . . . , n in the given order into the boxes along rows, starting with 1 in the (1, 1)-box and ending ... n T (λ) with n in the last box of the last row. Clearly, for any λ-tableau T , there is a unique s ∈ Sn with T = sT (λ). The minimal length of a product representing s in terms of the Coxeter generators s1 , . . . , sn−1 is called the length of s and denoted by `(s). It is a standard fact that `(s) equals the number of inversions of s, that is, the number of pairs (i, j) ∈ [n] × [n] with i < j but s(i) > s(j); see Exercise 4.17 or Example 7.10. 1 2 3

Lemma 4.8. (a) Let T be a λ-tableau. Then there is a finite sequence si1 , . . . , sil of admissible transpositions such that si1 . . . sil T = T (λ) and l = `(si1 . . . sil ). (b) Let c, c0 ∈ Cont(n). Then c ∼ c0 if and only if there is a finite sequence of admissible transpositions that transforms c into c0 . Proof. (a) In order to prove the existence of a sequence of admissible transpositions transforming T into T (λ), let nT be the number in the box at the right-hand end of the last row of T . We argue by induction on n and n − nT . The case n = 1 being trivial, assume that n > 1. If nT = n then remove the last box from T and let T 0 denote the resulting standard Young tableau, of shape λ0 ` n−1. By induction on n, we can transform T 0 into T (λ0 ) by a sequence of admissible transpositions given by Coxeter generators ∈ Sn−1 , and the sequence may be chosen to have the desired length. The same sequence will move T to T (λ). Now suppose that nT < n. Note that the box of T containing nT + 1 cannot occur in the same row or column as the last box, containing nT . Therefore, snT is an admissible transformation for T . The λ-tableau T 0 = snT T satisfies nT 0 = nT + 1. By induction, there is a finite sequence si1 , . . . , sil of admissible transpositions such that s = si1 . . . sil satisfies sT 0 = T (λ) and l = `(s). It follows that ssnT T = T (λ) and l+1 = `(ssnT ), where the latter equality holds because s(nT ) < n = s(nT +1) (Exercise 4.17). (b) Clearly, the existence of a sequence of admissible transpositions that transforms c into c0 implies that c and c0 differ only by a place permutation and so c ∼ c0 . The converse follows from (a) and (4.21). t u Description of Cont(n) The following proposition gives the desired description of the set Cont(n). Later in this section, we will compare Cont(n) with another set of n-tuples, Spec(n) ⊆ kn . For this, reason, we view Cont(n) ⊆ kn . Conditions (i) and (ii) below imply that, in fact, Cont(n) ⊆ Zn . Proposition 4.9. Cont(n) is precisely the set of all c = (c1 , c2 , . . . , cn ) ∈ kn satisfying the following conditions: (i) c1 = 0;

202

4 The Symmetric Groups

(ii) ci − 1 or ci + 1 ∈ {c1 , c2 , . . . , ci−1 } for all i ≥ 2; (iii) if ci = cj = a for i < j then {a + 1, a − 1} ⊆ {ci+1 , . . . , cj−1 }. Proof. Let C(n) denote the set of n-tuples c = (c1 , c2 , . . . , cn ) ∈ kn satisfying conditions (i) – (iii). We need to show that Cont(n) = C(n). We first check that Cont(n) ⊆ C(n). As we have observed earlier, condition (i) certainly holds if c = cT for some standard Young tableau T , because the number 1 must be in the (1, 1)-box; any i ≥ 2 must occupy a box of T in position (x, y) with x > 1 or y > 1. In the former case, let j be the entry in the (x − 1, y)-box. Then j < i and cj = y − (x − 1) = ci + 1, whence ci + 1 ∈ {c1 , c2 , . . . , ci−1 }. In an analogous fashion, one shows that ci − 1 ∈ {c1 , c2 , . . . , ci−1 } if y > 1, proving (ii). y i k l j

y =x+a

x

Now suppose that ci = cj = a for i < j. Then the entries i and j both lie on the adiagonal; say i occupies the (x, x + a)-box and j the (x0 , x0 + a)-box, with x < x0 . Let k and l denote the entries in the boxes at positions (x + 1, x + a) and (x0 − 1, x0 + a), respectively. Then k, l ∈ {i + 1, . . . , j − 1} and ck = (x + a) − (x + 1) = a − 1, cl = (x0 + a) − (x0 − 1) = a + 1. This proves (iii), thereby completing the proof of the inclusion Cont(n) ⊆ C(n).

For the reverse inclusion, Cont(n) ⊇ C(n), we proceed by induction on n. The case n = 1 being clear, with C(1) = {(0)} = Cont(1), assume that n > 1 and that C(n − 1) ⊆ Cont(n − 1). Let c = (c1 , c2 , . . . , cn ) ∈ C(n) be given. Clearly, the truncation c0 = (c1 , c2 , . . . , cn−1 ) also satisfies conditions (i) – (iii); so c0 ∈ C(n − 1) ⊆ Cont(n − 1). Therefore, there exists a (unique) standard Young tableau T 0 with content cT 0 = c0 . We wish to add a box containing the number n to T 0 so as to obtain a standard Young tableau, T , with cT = c. Thus, the new box must be placed on the cn -diagonal y = x + cn , at the first slot not occupied by any boxes of T 0 . We need to check that the resulting T has “flag shape” so that it represents a partition; the monotonicity requirement for standard Young tableaux is then automatic, because the new box contains the largest number. First assume that cn ∈ / {c1 , c2 , . . . , cn−1 }; so T 0 has no boxes on the cn -diagonal. Since there are no gaps between the diagonals of T 0 , the values c1 , . . . , cn−1 , with repetitions omitted, form an interval in Z containing 0. Therefore, if cn > 0 then cn > ci for all i < n, while condition (ii) tells us that cn − 1 ∈ {c1 , c2 , . . . , cn−1 }. Thus, cn = max{ci | 1 ≤ i ≤ n − 1} + 1 and the new box labeled n is added at the right end of the first row of T 0 , that is, at position (1, 1 + cn ). Similarly, if cn < 0, then the new box is added at the bottom end of the first column of T 0 . In either case, the resulting T has flag shape.

4.4 Proof of the Graph Isomorphism Theorem

203

y Finally assume that cn ∈ {c1 , c2 , . . . , cn−1 } and choose i < n maximal with ci = cn =: a. Then the box labeled i is the last box on the ai s diagonal of T 0 . We also know from condition (iii) r n that there exist r, s ∈ {i + 1, . . . , n − 1} with cr = a − 1 and cs = a + 1. In fact, r and s y =x+a are both unique. Indeed, if i < r < r0 < n and cr = a − 1 = cr0 , then (iii) implies that a ∈ x {cr+1 , . . . , cr0 −1 }, contradicting maximality of i. This shows uniqueness of r; the argument for s is analogous. Therefore, T 0 has unique boxes on the (a − 1)- and (a + 1)-diagonals with entries > i. Necessarily these boxes are the last ones on their respective diagonals, and they must be situated underneath and to the right of the ibox, respectively. Therefore, the new box labeled n is slotted in to the corner formed by the boxes with i, r and s, again resulting in the desired flag shape for the Young tableau T . t u

4.4.2 Weights Returning to the representation theoretic side of matters, we will now define another set of n-tuples, Spec(n). In §4.4.3, we will show that Spec(n) is in fact the same as Cont(n), but this will require some preparations. The definition of Spec(n) uses the Gelfand-Zetlin algebra GZ n . Recall that GZ n is a commutative split semisimple k-algebra (Theorem 4.4). Therefore, there is a bi∼ jection HomAlgk (GZ n , k) ←→ Spec GZ n = MaxSpec GZ n , φ 7→ Ker φ (§1.3.2). Moreover, each φ is determined by its values on the distinguished set of algebra generators of GZ n consisting of the JM-elements X1 , . . . , Xn (Corollary 4.2). Therefore, Spec GZ n is in bijection with the following set of n-tuples: def

Spec(n) =

 (φ(X1 ), φ(X2 ), . . . , φ(Xn )) | φ ∈ HomAlgk (GZ n , k) ⊆ kn

By (4.10) we further know that HomAlgk (GZ n , k) is in bijection with the set of all paths T : 1S1 → · · · → V in B terminating in some V ∈ Irr Sn . More precisely, the GZ-basis {vT } of each V consists of weight vectors for GZ n and HomAlgk (GZ n , k) consists of the corresponding weights φT : (a ∈ GZ n )

a.vT = φT (a)vT

(4.24)

Thus, we have bijections

1

via GZ-bases

n

paths 1S1 → . . . in B with endpoint ∈ Irr Sn

∈

αT := φT (Xi )


n

∼

∈

∈

Spec(n) ∼ HomAlgk (GZ n , k) φT

T

The first component of each αT is 0, because X1 = 0.

o (4.25)

204

4 The Symmetric Groups

Equivalence relation Define ≈ on Spec(n) by αT ≈ αT 0

def

⇐⇒ ⇐⇒

φT and φT 0 are weights of the same representation ∈ Irr Sn T and T 0 are paths 1S1 → . . . in B with the same endpoint ∈ Irr Sn

From (4.25) we obtain the following bijections: Spec(n)

n

∼ via GZ-bases

paths 1S1 → . . . in B with endpoint ∈ Irr Sn remember endpoint

Spec(n)/≈

∼

o

(4.26)

Irr Sn

Description of Spec(n) Elements of Spec(n) will simply be written as n-tuples, α = (a1 , a2 , . . . an ) ∈ kn We will denote the path T corresponding to α in (4.25) by Tα and the weight vector vTα by vα . Furthermore, V (α) will denote the endpoint of Tα ; so vα is a member of the GZ-basis of V (α) and V (α) is the unique V ∈ Irr Sn such that α is a weight of V . In this notation, we have α ≈ α0 if and only if V (α) = V (α0 ) and formula (4.24) becomes Xk .vα = ak vα for all k (4.27) The vector vα ∈ V (α) is determined by these equations up to a scalar multiple. We will scale the weight vectors vα in a consistent way in §4.5.1, but this will not be necessary for now. Example 4.10 (Spec(4)). We need to find the GZ-basis and weights of each V ∈ Irr S4 . For the representation 1, this is trivial. Note also that, for any n, the unique weight of 1 is (0, 1, 2, . . . , n − 1), because Xk .1 = (k − 1) . The GZ-bases of V3 and Ve2 were determined in Example 4.3. In the case of Ve2 , note that X4 acts on Ve2 via the canonical map kS4  kS3 , which sends X4 7→ (2, 3) + (1, 3) + (1, 2). Next, for any n and any V ∈ Irr Sn , the sign twist V ± = sgn ⊗V has the “same” GZ-basis as V but with weights multiplied by −1: X Xk .(1 ⊗ vα ) = − (i, k).(1 ⊗ vα ) = −(1 ⊗ Xk .vα ) = −ak (1 ⊗ vα ) i s(i + 1) Deduce that l(s) = `(s). 4.18 (Some decompositions into irreducibles). Let Vn−1 = V (n−1,1) be the standard representation of Sn and Mn the standard permutation representation. Prove: S ⊕2 (n−2,1,1) (a) Vn−1 ⊗ Mn ∼ ⊕ V (n−2,2) = 1↑Snn−2 ∼ = 1 ⊕ Vn−1 ⊕ V (n−2,1,1) (b) Vn−1 ⊗ Vn−1 ∼ ⊕ V (n−2,2) = 1 ⊕ Vn−1 ⊕ V

4.5 The Irreducible Representations The purpose of this section is to derive some explicit formulae for the action of Sn on certain well-chosen bases for the irreducible representations V λ . In particular, we shall see that each V λ has a GZ-basis so that the matrices of all operators sV λ (s ∈ Sn ) have entries in Q; for a different choice of normalization of the GZ-basis, the matrices will turn out to be orthogonal. Finally, we will present an efficient method for determining the irreducible characters χλ = χV λ (Murnaghan-Nakayama Rule). 4.5.1 Realization over Q Let λ ` n and let V λ be the corresponding irreducible representation of Sn as per the Graph Isomorphism Theorem. Since paths 1S1 → · · · → V λ in B are in bijection → · · · → λ in Y or, equivalently, λ-tableaux, with weights being with paths identified with contents, we can rewrite (4.6) in the form M Vλ = VTλ T a λ-tableau

with uniquely determined 1-dimensional subspaces VTλ . Specifically, VTλ is the GZ n weight space of V λ for the weight cT , the content of T . We will now select a nonzero vector from each VTλ in a coherent manner so as to obtain a special GZ-basis of V λ . To this end, let πT : V λ  VTλ denote the projection along the sum of the weight spaces VTλ0 with T 0 6= T and fix 0 6= v(λ) ∈ VTλ(λ)

210

4 The Symmetric Groups

Here T (λ) is the λ-tableau considered in Lemma 4.8. Each λ-tableau T has the form T = sT T (λ) for a unique sT ∈ Sn . Put
vT := πT sT .v(λ) (4.30) The check that all vT are nonzero is done in the theorem below. More importantly, the theorem shows that the action of Sn on the resulting GZ-basis of V λ is defined over Q. Adopting the notation of Proposition 4.11, we will write dT,i := (cT,i+1 − cT,i )−1

(4.31)

where cT = (cT,1 , . . . , cT,n ) is the the content of T . Thus, dT,i is a nonzero rational number of absolute value ≤ 1. Recall also from (4.23) that dT,i 6= ±1 if and only if the Coxeter generator si ∈ Sn is admissible for T , that is, si T is a λ-tableau. Theorem 4.15. Let λ ` n . For each λ-tableau T , let vT and dT,i be defined as in (4.30), (4.31). Then {vT } is a GZ-basis of V λ and the action of si = (i, i + 1) on this basis is as follows: (i) if dT,i = ±1 then si .vT = dT,i vT ; (ii) if dT,i 6= ±1 then ( dT,i vT + vsi T si .vT = dT,i vT + (1 − d2T,i ) vsi T

−1 if s−1 T (i) < sT (i + 1) −1 if s−1 T (i) > sT (i + 1)

Proof. Proposition 4.11 in conjunction with (4.21), (4.23) implies that si .VTλ ⊆ VTλ if si is not an admissible transposition for T and si .VTλ ⊆ Vsλi T + VTλ if si is an λ = admissible transposition for T . Consequently, putting `(T ) = `(sT ) and V