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The Invariant Theory of Matrices
 147044187X, 9781470441876

Table of contents :
Cover
Title page
Table of Contents
protect oindent Introduction and preliminaries
1. Sectionformat {Introduction}{1}
2. Sectionformat {Preliminaries}{1}
Part I . protect enspace protect oindent The classical theory
3. Sectionformat {Representation theory}{1}
4. Sectionformat {Algebras with trace}{1}
Part II . protect enspace protect oindent Quasi-hereditary algebras
5. Sectionformat {Modules}{1}
6. Sectionformat {Good filtrations and quasi-hereditary algebras}{1}
Part III . protect enspace protect oindent The Schur algebra
7. Sectionformat {The Schur algebra}{1}
8. Sectionformat {Double tableaux}{1}
9. Sectionformat {Modules for the Schur algebra}{1}
10. Sectionformat {Rational $GL(m)$-modules}{1}
11. Sectionformat {Tensor products}{1}
Part IV . protect enspace protect oindent Matrix functions and invariants
12. Sectionformat {A reduction for invariants of several matrices }{1}
13. Sectionformat {Polarization and specialization}{1}
14. Sectionformat {Exterior products}{1}
15. Sectionformat {Matrix functions and invariants}{1}
Part V . protect enspace protect oindent Relations
16. Sectionformat {Relations}{1}
17. Sectionformat {Describing $K_m$}{1}
18. Sectionformat {$K_m$ versus $ ilde K_m$}{1}
Part VI . protect enspace protect oindent The Schur algebra of~a~free~algebra
19. Sectionformat {Preliminary facts}{1}
20. Sectionformat {The Schur algebra of the free algebra}{1}
Bibliography
General Index
Symbol Index
Back Cover

Citation preview

UNIVERSITY LECTURE SERIES VOLUME 69

The Invariant Theory of Matrices Corrado De Concini Claudio Procesi

American Mathematical Society

The Invariant Theory of Matrices

UNIVERSITY LECTURE SERIES VOLUME 69

The Invariant Theory of Matrices Corrado De Concini Claudio Procesi

American Mathematical Society Providence, Rhode Island

EDITORIAL COMMITTEE Jordan S. Ellenberg William P. Minicozzi II (Chair)

Robert Guralnick Tatiana Toro

2010 Mathematics Subject Classification. Primary 15A72, 14L99, 20G20, 20G05.

For additional information and updates on this book, visit www.ams.org/bookpages/ulect-69

Library of Congress Cataloging-in-Publication Data Names: De Concini, Corrado, author. | Procesi, Claudio, author. Title: The invariant theory of matrices / Corrado De Concini, Claudio Procesi. Description: Providence, Rhode Island : American Mathematical Society, [2017] | Series: University lecture series ; volume 69 | Includes bibliographical references and index. Identifiers: LCCN 2017041943 | ISBN 9781470441876 (alk. paper) Subjects: LCSH: Matrices. | Invariants. | AMS: Linear and multilinear algebra; matrix theory – Basic linear algebra – Vector and tensor algebra, theory of invariants. msc | Algebraic geometry – Algebraic groups – None of the above, but in this section. msc | Group theory and generalizations – Linear algebraic groups and related topics – Linear algebraic groups over the reals, the complexes, the quaternions. msc | Group theory and generalizations – Linear algebraic groups and related topics – Representation theory. msc Classification: LCC QA188 .D425 2017 | DDC 512.9/434–dc23 LC record available at https://lccn. loc.gov/2017041943

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2017 by the American Mathematical Society. All rights reserved.  The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines 

established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

22 21 20 19 18 17

Table of Contents Introduction and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 8

Part I. The classical theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3. Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4. Algebras with trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Part II. Quasi-hereditary algebras . . . . . . . . . . . . . . . . . . . . . . 39 5. Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6. Good filtrations and quasi-hereditary algebras . . . . . . . . . . . . . . . . . . 43 Part III. The Schur algebra . . . . . 7. The Schur algebra . . . . . . . . . . . . 8. Double tableaux . . . . . . . . . . . . . 9. Modules for the Schur algebra . . . 10. Rational GL(m)-modules . . . . . . 11. Tensor products . . . . . . . . . . . .

....... ........ ........ ........ ........ ........

. .. .. .. .. ..

49 50 51 62 75 78

Part IV. Matrix functions and invariants . . . . . . . . . 12. A reduction for invariants of several matrices . . . . . . . 13. Polarization and specialization . . . . . . . . . . . . . . . . . . 14. Exterior products . . . . . . . . . . . . . . . . . . . . . . . . . . . 15. Matrix functions and invariants . . . . . . . . . . . . . . . . .

...... ....... ....... ....... .......

. . . . .

. .. .. .. ..

87 88 91 95 99

Part V. Relations . . . . . . 16. Relations . . . . . . . . . . 17. Describing Km . . . . . . ˜m . . . . . . 18. Km versus K

....... ........ ........ ........

. .. .. ..

107 108 110 118

...... ....... ....... .......

. . . . . .

. . . .

...... ....... ....... ....... ....... .......

...... ....... ....... .......

....... ........ ........ ........ ........ ........

....... ........ ........ ........

Part VI. The Schur algebra of a free algebra . . . . . . . . . . . . . . . 131 19. Preliminary facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 20. The Schur algebra of the free algebra . . . . . . . . . . . . . . . . . . . . . . . 135 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 General Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

v

Introduction and preliminaries

2

INTRODUCTION AND PRELIMINARIES

1. Introduction The purpose of these notes is to give a unified, complete and self contained exposition of the main algebraic theorems of invariant theory for matrices in a characteristic free approach. We only treat the general symbolic theory and do not discuss the several special theorems and computations which can be found in a comprehensive paper of Vesselin Drensky, [23], which also contains a rather complete literature. Nor do we treat the applications to non-commutative algebra which will appear in a forthcoming book Rings with polynomial identities and finite dimensional representations of algebras by E. Aljadeff, A. Giambruno, C. Procesi, A. Regev ([1]). In particular we will avoid the use of algebraic geometry, and results on the cohomology of line bundles over the flag variety which are the standard approach to the theory, see [34]. Instead, we shall use the theory of quasi-hereditary algebras and of standard bitableaux, an algebraic and combinatorial approach. If G is a group acting linearly on a finite dimensional vector space W , then G acts on the algebra S[W ∗ ] of polynomial functions on W by (gf )(w) := f (g −1 w) and one defines. Definition 1.1. The algebra of invariants of G acting on W is: S[W ∗ ]G := {f ∈ S[W ∗ ] | gf = f, ∀g ∈ G}. Invariant theory has a long and complicated history whose discussion goes far beyond the purpose of these notes. The prototype theory is the theory of symmetric functions, largely studied in the last two centuries. This can be formulated in an arithmetic way by taking the ring Z[x1 , . . . , xd ] of polynomials in d variables over the integers and the action of the symmetric group Sd permuting the variables. In this case the ring of invariants Z[x1 , . . . , xd ]Sd is a polynomial ring Z[σ1 (x), . . . , σd (x)] where the polynomials σi (x1 , . . . , xd ), called elementary symmetric functions are defined by the formula (1)

d 

(λ − xi ) = λd − σ1 (x)λd−1 + σ2 (x)λd−2 + . . . + (−1)d σd (x).

i=1

For a modern treatment see [41]. It is useful to pass to infinitely many variables as follows, consider the map πd+1 : Z[x1 , . . . , xd , xd+1 ] → Z[x1 , . . . , xd ] given by πd+1 : xd+1 = 0. Then the restriction of πd+1 to Z[x1 , . . . , xd , xd+1 ]Sd+1 = Z[σ1 , . . . , σd , σd+1 ] maps it to Z[x1 , . . . , xd ]Sd and is given by πd+1 : σd+1 = 0. Thus one can define a limit algebra, called the ring of symmetric functions which is the polynomial ring in the infinitely many variables σi (x), i ∈ N. The ring Z[x1 , . . . , xd ]Sd is obtained from this ring by setting σi (x) = 0, ∀i > d. Identities among different types of symmetric functions are usually described in the limit ring. In the last century, due to work of Brauer, Schur and Weyl invariant theory is tied to representation theory. We will sketch some of the theory in §3.1. In particular in the book of H. Weyl The classical groups [72], it is developed a first fundamental theorem, FFT and a second fundamental theorem, SFT for invariants of several copies of the defining representation V of GL(V ) and its dual V ∗ and

1. INTRODUCTION

3

also for the other classical groups, in all these cases tensor symmetry is the main tool. These notes deal with a more difficult case in which a first and a second fundamental theorems can be proved, the case of several copies of End(V ) under conjugation action by GL(V ). Moreover contrary to Weyl’s approach which is valid only in characteristic 0 we will develop a characteristic free approach. We will not discuss the theory of several copies of End(V ) under conjugation action by the classical groups O(V ) or Sp(V ) when V has a non degenerate symmetric, resp. skew symmetric bilinear form. Such a theory exists in characteristic 0, see [45] or [48], and in a characteristic free approach one can see [77] and [39]. Thus the object of study is the following. Let F be an infinite field or the integers Z, we will denote by Mm (F ) the algebra of m×m matrices with coefficients in F . Given two integers n, m consider the ring (2)

Sm [ξ1 , . . . , ξn ] = F [xhi,j ], i, j = 1, . . . , m; h = 1, . . . , n

the (commutative) algebra of polynomial functions (in m2 n variables) on the vector space Mm (F )n of n-tuples of m × m matrices. The symbols ξh refer to the generic matrices ξh := (xhi,j ). The linear group GL(m, F ) acts on the space Mm (F )n by simultaneous conjugation. So the algebra Sm [ξ1 , . . . , ξn ]GL(m,F ) of invariants of n-tuples of m × m matrices, is the subring of polynomials f such that f (gξ1 g −1 , . . . , gξn g −1 ) = f (ξ1 , . . . , ξn ), ∀g ∈ GL(m, F ). Remark 1.2. As we shall see this ring of invariants is in fact defined over the prime field, and obtained by change of scalars from the invariants defined over Z. The question is to describe Sm [ξ1 , . . . , ξn ]GL(m,F ) . In classical language (as in Weyl’s book [72]) one describes the invariants of a given representation by a First and second fundamental theorem • The first fundamental theorem, FFT consists in describing a set of generators for the invariants. • The second fundamental theorem, SFT consists in describing the relations between the given set of generators for the invariants. In principle one may also try to study the syzygies but these are usually too hard to describe (but one has in this direction, at least one important result due to Lascoux [37] in characteristic 0). Let us start from just one (generic) matrix ξ = (xi,j ), i, j = 1, . . . m. In the algebra S[ξ] = F [xi,j ], of polynomial functions on Mm (F ), consider the characteristic polynomial of ξ: (3)

det(λ − ξ) = λm − σ1 (ξ)λm−1 + σ2 (ξ)λm−2 + . . . + (−1)m σm (ξ).

Since det(ξ) = det(gξg −1 ) the determinant (3) is an invariant function and then, also all its coefficients σi (ξ) ∈ S[ξ] are invariant under conjugation. Here σ1 (ξ) = tr(ξ) is the trace and σm (ξ) = det(ξ) is the determinant. A first basic theorem is Theorem 1.3. The ring of polynomial invariants of a matrix ξ under conjugation is freely generated by the m functions σi (ξ). (4)

S[ξ]GL(m) = F [σ1 (ξ), . . . , σm (ξ)].

4

INTRODUCTION AND PRELIMINARIES

In fact this is usually proved by a classical method which has far reaching generalizations (due to Chevalley), the restriction Theorem to diagonal matrices. Inside Mm (F ) we have the diagonal matrices which we identify to F m and inside GL(m, F ) we have the symmetric group Sm , of permutation matrices, which under conjugation induces the permutation action on F m . It follows that, if we restrict a G invariant function f on Mm (F ) to F m , we obtain a symmetric function. If x = diag(x1 , . . . , xm ) is a diagonal matrix with entries xi its characteristic polynomial is given by formula (1), therefore the coefficients of the characteristic polynomial restrict (up to the sign) to the elementary symmetric functions σi (x1 , . . . , xm ). When F is an infinite field we have Theorem 1.4. If F is an infinite field, the restriction of the ring of G invariant ∼ = polynomials on Mm (F ) to F m , S[ξ]GL(m) −→ F [x1 , . . . , xm ]Sm is an isomorphism with the ring of symmetric polynomials F [x1 , . . . , xm ]Sm = F [σ1 (x), . . . , σm (x)]. Proof. The restriction map is surjective since the elementary symmetric functions generate the ring of symmetric functions. So it is enough to show that it is injective. Since the field is infinite we may assume that it is algebraically closed. Then consider the subset U of Mm (F ) of matrices with distinct eigenvalues, this is the affine open set where the discriminant is nonzero. The discriminant is a polynomial with integer coefficients in the coefficients of the characteristic polynomial. If f is an invariant polynomial which restricted to the diagonal matrices is zero then it is also zero when restricted to U since every element of U is conjugate to a diagonal matrix. But U is Zariski open non empty, hence f = 0.  If F = Q the ring of symmetric functions is also generated by the Newton functions m  xki , ψk (ξ) = tr(ξ k ), k = 1, . . . , m. ψk (x1 , . . . , xm ) := i=1

There are universal formulas (cf. (21)) expressing the elements σi (ξ) as polynomials with rational coefficients in the elements tr(ξ k ), and conversely, the elements tr(ξ k ) as polynomials with integer coefficients in the elements σi (ξ). Remark 1.5. From the theory of symmetric functions it also follows that there are universal formulas expressing the elements σi (ξ j ) as polynomials with integer coefficients in the elements σh (ξ). 1.0.1. The theorems over Z or over any field F for any number of m × m matrices. We state now the Theorems which will be proved in this lecture note. Their proof is quite different in characteristic 0, where one can follow a very classical approach and in positive characteristic or over the integers where things are extremely more complicated. Take a finite set X = {x1 , . . . , xn } which we call an alphabet (later on we are going to allow ourselves to even take X infinite countable). Denote by F X = F x1 , . . . , xn  the free algebra with basis over F all the words in the alphabet X, also called monomials. Definition 1.6. We say that a monomial of positive degree (or a non empty word) w is primitive if it is not of the form w = w0k , k > 1. Let Wp denote the set of primitive monomials.

1. INTRODUCTION

5

Now consider cyclic equivalence of monomials that is Definition 1.7. We say that two monomials w1 , w2 are cyclically equivalent, c and we write w1 ∼ w2 if w1 = ab, w2 = ba for some monomials a, b. Lemma 1.8. If w is primitive we have exactly (w) distinct monomials equivalent to wk for all k ≥ 1. c If w1 , w2 are both primitive and w1k1 ∼ w2k2 are cyclically equivalent we have c k1 = k2 and w1 ∼ w2 are cyclically equivalent. Proof. This is an easy exercise left to the reader.



Remark 1.9. In a class of cyclically equivalent primitive words there is a unique minimum word in lexicographic order, this is called a Lyndon word (cf. [40]). Let W0 denote a set of representatives of primitive monomials up to cyclic equivalence (for instance we can choose the Lyndon words). We order W0 by degree and then lexicographically. By the universal property the free algebra can be evaluated in any associative algebra and in particular evaluates on m × m matrices ξi to matrix valued polynomial functions. Then, given any f (x1 , . . . , xn ) ∈ F x1 , . . . , xn , the coefficients σi (f (ξ1 , . . . , ξn )) are clearly invariants. We then have: Theorem 1.10 (FFT). The ring of invariants Sm [ξ1 , . . . , ξn ]GL(m) of n-tuples of matrices is generated by the elements σi (M ), i = 1, . . . , m with M any primitive monomial in ξ1 , . . . , ξn . If F has characteristic 0, the ring of invariants of n-tuples of matrices is also generated by the elements tr(M ) with M any monomial in ξ1 , . . . , ξn . This Theorem, in characteristic 0 is classical and appears in the papers of C. Procesi [45] and K. S. Sibirskii [59]. In positive characteristic it is due to S. Donkin [20], [21] as a consequence of a general theory which will be presented in §7 and of some identities to be presented in §14. The proof of this theorem in all characteristics is given in section 15.2. Remark 1.11. For each i and matrices A, B we have σi (AB) = σi (BA) that is σi (M ) depends on M only up to cyclic equivalence (cf. Definition 1.7). Proof. In fact since this is a formal identity it is enough to prove it for A invertible and then AB = A(BA)A−1 and the statement follows from the fact that  σi is conjugation invariant. For the second fundamental theorem one needs a substitutional calculus on the polynomial ring in the infinitely many variables: S = F [σi (p)], i = 1, . . . , ∞, p ∈ W0 . Let us denote by F+ X the free algebra without 1. One defines (cf. Proposition 4.24) degree i polynomial maps σi : F+ X → S = F [σi (p)] which on a given primitive monomial p coincide with σi (p). Theorem 1.12 (SFT). The ring of invariants of n-tuples of m × m matrices is the quotient of the formal ring S = F [σi (p)], i = 1, . . . , ∞, p ∈ W0 modulo the ideal generated by all values σi (f ), ∀i > m, ∀f ∈ F+ X (here F is an infinite field).

6

INTRODUCTION AND PRELIMINARIES

This Theorem, in characteristic 0 is due independently to C. Procesi [45] and Y. Razmyslov [50]. In positive characteristic it is due to A. Zubkov [75]. Since σi is a polynomial map of degree i, it gives rise under polarization to maps σi1 ,...,ik (M1 , . . . , Mk ), defined for all choices of indices (i1 , . . . ik ) and M1 , . . . Mk , belonging to the natural basis W of monomials of F+ X. The Theorem also holds for F any field  or F = Z taking the ideal generated by all values σi1 ,...,ik (M1 , . . . , Mk ), with j ij > m. Finally, Ziplies [73], [74] discovered a very compact and surprising formulation of the previous Theorems which he proved in characteristic 0 and to which Vaccarino [69] gave its final form. In these notes we show that the following Theorem is valid whenever F is any field or the integers: Theorem 1.13. (1) The map det : F x1 , . . . , xn  → Sm [ξ1 , . . . , ξn ]GL(m) , f (x1 , . . . , xn ) → det(f (ξ1 , . . . , ξn )) is multiplicative. (2) The induced map det : (F x1 , . . . , xn ⊗m )Sm → Sm [ξ1 , . . . , ξn ]GL(m) is surjective (FFT). (3) Its kernel is the ideal generated by all commutators (SFT). Finally one has a non commutative variation of the theorems. One considers the algebra Pm,n of GL(m) equivariant polynomial maps f : Mm (F )n → Mm (F ): f (gξ1 g −1 , . . . , gξn g −1 ) = gf (ξ1 , . . . , ξn )g −1 ,

∀g ∈ GL(m, F ).

Then the coordinate maps ξi : (ξ1 , . . . , ξn ) → ξi are among these maps as well as the invariants (since F · 1m ⊂ Mm (F )). One has Theorem 1.14. (1) The non commutative algebra Pm,n of equivariant polynomial maps is generated by the coordinates ξi over the ring of invariants. (2) Every relation in this algebra can be deduced from the Cayley–Hamilton identities CHk (ξ) := ξ k − σ1 (ξ)ξ k−1 + σ2 (ξ)ξ k−2 + . . . + (−1)k σk (ξ), ∀k ≥ m. (3) In characteristic 0 it is enough to consider CHm (ξ). (4) If F is finite, or Z, one has to replace in 2) the polynomial  CHk (ξ) with hi ≥ m. all its possible polarized forms CHh1 ,...,hr (M1 , . . . , Mr ), 1.0.2. The plan of these notes. These notes are organized into 6 Parts. In the first we prove the Theorems in characteristic 0. In this case only basic methods of representation theory are used, in particular tensor symmetry and Schur–Weyl duality. In Part 2 we introduce the notion of quasi hereditary algebras due to Cline, E. ; Parshall, B. ; Scott, L., [9], [10], [11], [12], [13] and explain some of the properties of their representations, in particular we introduce the notion of good filtration which plays a central role in our notes. In Part 3 we study a particular but very important class of quasi hereditary algebras, that of Schur algebras, introduced by Green in [27]. This allows us to introduce the notion of representations of the group GL(m, F ) having a good filtrations. This is a special case of a more general theory which holds for arbitrary reductive algebraic groups, see the book of Donkin [19]. In particular the Theorem

1. INTRODUCTION

7

that the tensor product of two modules with good filtration, here proved in a combinatorial way 11.1, has a good filtration is due to Wang [71] in this case and the final result for general reductive groups to Olivier Mathieu [42] using the deep ideas of Frobenius splittings by Mehta and Ramanathan [43]. This is based on results of the cohomology of line bundles on the flag variety, which are beyond the purpose of these notes for which we refer to the book [34] by Jantzen. As mentioned above we develop this theory, by combinatorial methods, only for the linear group, thus avoiding all the deep problems of algebraic geometry and homological algebra. This suffices for our purposes. The reader who is familiar with the work of Donkin and others on good filtrations may want to skip this part. In Part 4 we develop the special tools needed to apply the theory of good filtrations to our situation and we prove the First Fundamental Theorem. Part 5 is devoted to the work of Zubkov and the Second Fundamental Theorem. Finally Part 6 develops the notion of the Schur algebras of a free algebra and then explains the theorem of Ziplies and Vaccarino in a characteristic free way. Finally we should point out that, especially in dimension 3 and for the orthogonal group there is an extensive literature of matrix invariants in continuum physics and elasticity theory, mostly by Anthony James Merrill Spencer and Ronald Rivlin, see [61], [62], [63], [64], [65].

8

INTRODUCTION AND PRELIMINARIES

2. Preliminaries In this section we establish some notations and collect a few preliminary, and standard, facts which we are going to use in the notes. 2.1. Multilinear algebra. Let us recall some basic notations and facts. Given a module M over a commutative ring A one has the tensor algebra ⊗k T (M ) := ⊕∞ . Denote by i : M = M ⊗1 → T (M ) the inclusion. k=0 M T (M ) has the universal property that any linear map f : M → R with R an A-algebra extends to a unique homomorphism of A-algebras f¯ : T (M ) → R making the diagram / T (M ) MF FF FFf ¯ FF FF f "  R i

commutative. The quotient of T (M ) by the ideal generated by all commutators m1 ⊗ m2 − k m2 ⊗ m1 is the symmetric algebra S(M ) := ⊕∞ k=0 S (M ). S(M ) has the same universal property but now for commutative algebras R. If M is free of rank h with basis e1 , . . . , eh then S(M ) is identified with the ring of polynomials in k variables, A[e1 , . . . , eh ].  k Another canonical quotient of T (M ) is the exterior algebra M := ⊕∞ M k=0 quotient of T (M ) by the ideal generated by the elements m⊗2 , m ∈ M . The multiplication is denoted by ∧. If M is free with basis e1 , . . . , eh then    M := ⊕hk=0 k M and a basis of k M is formed by the elements ej1 ∧ . . . ∧ ejk , 1 ≤ j1 < j2 < . . . < jk ≤ h. Remark 2.1. For a finite dimensional vector space M over a field F , or just a  finite free module over a commutative ring A one has an identification of ( k M )∗ k ∗ with M given by the pairing:1 φ1 ∧ . . . ∧ φk | v1 ∧ . . . ∧ vk  := det(φi | vj ), φi ∈ M ∗ , vj ∈ M.

(5)

Under this pairing we see that the dual basis of the basis ej1 ∧ ej2 ∧ . . . ∧ ejk is the basis ej1 ∧ ej2 ∧ . . . ∧ ejk . Where e1 , . . . , eh is the basis of M ∗ dual to e1 , . . . , eh . Assuming that M is free of rank h over A, we have a non degenerate pairing p: Since

h

k 

M⊗

h−k 

M −→

h 

M,

p(a ⊗ b) := a ∧ b, 0 ≤ k ≤ h.

M is free of rank 1, this gives a natural isomorphism

h−k h h−k h     M) = ( M )∗ ⊗ M = M∗ ⊗ M. h Given a basis e1 , . . . , eh of M one has M = Au, u := e1 ∧ . . . ∧ eh and

p:

k 

M = hom(

h−k 

M,

h 

p(ej1 ∧ . . . ∧ ejk )(ei1 ∧ . . . ∧ eih−k ) = ej1 ∧ . . . ∧ ejk ∧ ei1 ∧ . . . ∧ eih−k 1 We use the bracket notation of physics φ | v to denote the evaluation of a linear form φ on a vector v.

2. PRELIMINARIES

9

which is either equal to 0 or to ε e1 ∧ . . . ∧ em where ε = ±1 is the sign of the permutation j1 , . . . , jk , i1 , . . . , ih−k . Since {j1 , . . . , jk } is the complement of {i1 , . . . , ih−k } this sign is determined by these elements and can be denoted by εj1 ,...,jk . Thus (6)

p(ej1 ∧ . . . ∧ ejk ) = εj1 ,...,jk ei1 ∧ . . . ∧ eih−k ⊗ u, 2.2. Polynomial maps. In M

⊗k

u = e1 ∧ . . . ∧ eh .

we have the action of the symmetric group

Sk : (7)

σ(m1 ⊗ . . . ⊗ mk ) = mσ−1 (1) ⊗ . . . ⊗ mσ−1 (k)

and the space of symmetric tensors (8)

(M ⊗k )Sk := {u ∈ M ⊗k | σ(u) = u, ∀σ ∈ Sk }. If M is a finite free module, the pairing M ∗⊗k ⊗ M ⊗k → A given by φ1 ⊗ . . . ⊗ φk | v1 ⊗ . . . ⊗ vk  :=

k 

φi | vi , φi ∈ V ∗ , vi ∈ V,

i=1

induces a perfect duality between S k (M ∗ ) and (M ⊗k )Sk . Furthermore M ⊗k is equipped with the homogeneous polynomial map m → m⊗k which is universal for such maps (see Proposition 2.6). This is in fact part of a general theory of polynomial laws which we just recall for convenience of the reader. The general theory of polynomial laws has been developed by Roby in [54] for modules over a commutative ring A. Let CA be the category of commutative Aalgebras. Given a A-module M we may consider the functor FM : Ca → Sets given by B → B ⊗A M Definition 2.2. A polynomial law f between two A-modules M, N is a natural transformation from the functor FM to the functor FN . This means that, for every commutative algebra B we have a set theoretic map fB : B ⊗A M → B ⊗A N , and for each homomorphism h : B → C of commutative algebras, we have the commutative diagram: fB

B ⊗A M −−−−→ B ⊗A N ⏐ ⏐ ⏐h⊗1 ⏐ h⊗1M   N. C ⊗A M −−−−→ C ⊗A N fC

For example a A-linear homomorphism f : M → N clearly defines the polynomial law fB = id ⊗ f : B ⊗A M → B ⊗A N . This is called a linear polynomial law. We denote by PA (M, N ) the polynomial laws from M to N . As polynomials, polynomial laws have the notion of degree. Definition 2.3. Let f ∈ PA (M, N ) be a polynomial law between A-modules M, N . We say that f is homogenous of degree d ≥ 0 if for all A-algebras B, b ∈ B, and z ∈ B ⊗A M , fB (bz) = bd fB (x). In what follows we are going to be interested only in the case in which M, N are two free modules of finite or countable rank. We denote by {d1 , d2 , . . . , dj , . . .} (resp. {e1 , e2 , . . . , ei , . . .}) a chosen basis of N (resp. M ). If this is the case and A is an infinite domain, a polynomial law fB : B ⊗A M → B ⊗A N is also called a polynomial map, and it is determined by a single map

10

INTRODUCTION AND PRELIMINARIES

f : M → N which, written in coordinates, has the property that, each coordinate of f (x), in the basis dj of N , is a polynomial in the coordinates of x in the basis ei of M .   (9) f ( xi ei ) = fj (x1 , . . . , xh , . . .)dj , fj (x1 , . . . , xh , . . .) ∈ A[x1 , . . . , xj , . . .]. i

j

Clearly the polynomial law f is homogeneous of degree d if and only if each polynomial fj is homogeneous of degree d. In our treatment we shall really only need this case. If A is a finite field, then by definition a polynomial law gives a map E ⊗A M → E ⊗A N for any field E ⊃ F, so it is still given by Formula (9). By abuse of notations we still denote it by f : M → N , although this function does not determine the law. Fix a free module M with a countable or finite basis {e1 , e2 , . . . , ei , . . .} over an infinite domain A and an integer m. Let us analyze the space of symmetric tensors (M ⊗m )Sm defined in formula (8). The symmetric group permutes the basis elements ej1 ⊗ · · · ⊗ ejm of M ⊗m . For any sequence I = i1 ≤ i2 ≤ · · · ≤ im consider the set  OI of all distinct sequences obtained by permuting I, an Sm orbit. We set eI = (j1 ,j2 ,··· ,jm )∈OI ej1 ⊗· · ·⊗ejm . Sometimes it is more convenient to describe the sequence I by the multiplicity αj with which the index  j appears in I, the corresponding function on N will be denoted by α we have |α| = i αi = m. Then the element eI will also be denoted by eα .

m α Observe that the stabilizer in Sm of e is j Sαj so that there are α1 ,...,αj ,... distinct elements in the orbit OI . Lemma 2.4. The A-module (M ⊗m )Sm is free with a basis consisting of the elements eI = eα , |I| = m. Let ui be a sequence of elements in A which are all zero except for a finite number. (10) (





ui ei )⊗m =

|I|=m

i



uI eI =

uα eα ,

m 

uI :=

α | |α|=m

uih = uα :=



i uα i .

i

h=1

Proof. This is clear since the OI are the orbits of Sm acting on a basis of  M ⊗m . Remark 2.5. In general, given any sequence a := (a1 , . . . , ak ) of elements ai ∈ M we may define, as a variation of Formula (10), the elements aα ∈ (M ⊗m )Sm  where α = (α1 , . . . , αk ), αi ∈ N and m = ki=1 αi , by introducing variables ti : (11)

(

k 

ti ai )⊗m =



tα a α , tα =

|α|=m

i=1

k 

i tα i .

i=1

The previous definition applies to any list, in particular we may have that the list  a is formed by r elementsbj each appearing hj times in the list, so that k = rj=1 hj . Then, setting sj = i | ai =bj ti : (

 i

ti ai )⊗m = (

    ( ti )bj )⊗m = ( sj bj )⊗m = sβ bβ , j

i | ai =bj

j

|β|=m

2. PRELIMINARIES

11

we then see that (12)

sβ =

k 

(



ti )βj =



j=1 i | ai =bj

cα,β tα , cα,β ∈ N,

α

aα =



cα,β bβ .

β

Proposition 2.6. The map i : M → (M ⊗m )Sm ,

(13)

given by i(a) = a⊗m , is a homogeneous polynomial map of degree m and satisfies the following universal property: Any polynomial map f : M → N , homogeneous of degree m between the free modules M and N factors uniquely through a linear map f¯ / (M ⊗m )Sm . MI II II II f¯ II f II  $ N  Proof. Formula (10), tells us that for a = i ui ei , we have   ui ei )⊗m = uI eI , i(a) = a⊗m = ( i

|I|=m

i

so i is clearly a polynomial map homogeneous of degree m. We verify directly the universal property. By definition, a  polynomial map, homogeneous of degree m from M to N is given by a polynomial |I|=m nI xI with values in N . Thus, since the elements eI are a linear basis of (M ⊗m )Sm , assigning the values nI := f¯(eI ) ∈ N we get a linear map f¯ : (M ⊗m )Sm → N and    f( ui ei ) = uI nI = f¯(( ui ei )⊗m ).  i

h

i

Corollary 2.7. If A = F is an infinite field the elements x⊗m span linearly M ⊗m . Proof. Otherwise there is a non zero linear map f : M ⊗m → F which vanishes on all the elements x⊗m . But such a linear map gives rise to a non zero polynomial of x a contradiction.  2.2.1. Multiplicative polynomial maps (According to Roby [55]). Let us consider now polynomial maps between algebras. Definition 2.8. If M, N, are both algebras one says that the polynomial map f : M → N is multiplicative if f (ab) = f (a)f (b), ∀a, b ∈ M. Notice that if M is an algebra then also (M ⊗m )Sm is an algebra and the map i : a → a⊗m , of Formula (13), is multiplicative. Definition 2.9. If R is an algebra then the algebra (R⊗m )Sm will be denoted by S m (R) and called the Schur algebra of R of degree m.

12

INTRODUCTION AND PRELIMINARIES

In fact the main example for these notes is the case where R = End(V ) (cf. §7.1) so that (End(V )⊗m )Sm = EndSm (V ⊗m ). Observe that, if R is graded then also R⊗m and (R⊗m )Sm are naturally graded. The basic property of multiplicative maps is: Proposition 2.10. Let M and N be algebras and f : M → N a multiplicative polynomial map homogeneous of degree m. Then the linear map f¯ : (M ⊗m )Sm → N is a homomorphism of algebras. Proof. One may assume that we work over an infinite field, in this case the  elements xm , x ∈ M span M ⊗m , by Corollary 2.7, so the claim is clear. Let us give a second basic example. Example 2.11. Take M = F [t] and N = S[ξ]GL(m,F ) the space of polynomial invariant functions on Mm (F ) (as usual we think of ξ as the generic m × m matrix). Consider the polynomial map det : M → N : det : F [t] → S[ξ]GL(m,F ) , f (t) → det(f (ξ)). The map det is multiplicative and homogeneous of degree m. From Proposition 2.10, we then see that the map det : F [t] → S[ξ]GL(m,F ) factors through a homomorphism det : (F [t]⊗m )Sm → S[ξ]GL(m,F ) . The algebra F [t]⊗m = F [t1 , . . . , tm ] hence S m (F [t]) = (F [t]⊗m )Sm is by definition the algebra of symmetric polynomials in m variables and we see that det inverts naturally the restriction map to diagonal matrices. Observe that the element (λ−t)⊗m ∈ (F [t]⊗m )Sm coincides with the polynomial of formula (1) defining the elementary symmetric functions σi (i.e. the element σm;i (t) ∈ (F [t]⊗m )Sm of Formula (103) coincides with the elementary symmetric function σi ). 2.3. Linear algebraic groups. We review very quickly some basic (and elementary) facts on linear algebraic groups. For a systematic approach the reader may consult [6], [34], [66].2 Let us assume that F is an algebraically closed field. The general linear group 2 GL(m, F ) is the basic linear algebraic group. It is the affine subvariety of F m +1 = Mm (F ) × F defined by the pairs (A, u) | det(A)u = 1. Thus its coordinate ring, as affine variety, is the ring of polynomials functions on m × m matrices with the determinant inverted, F [xi,j ][det−1 ]. The center of GL(m, F ) consists of scalar matrices and it is isomorphic to the multiplicative group F ∗ . Definition 2.12. A linear algebraic group G over the field F is a subgroup G ⊂ GL(m, F ), for some m, which is also an algebraic subvariety. The restriction of the ring of functions F [xi,j ][det−1 ] to G is the coordinate ring of G or the ring of regular functions of G, denoted A[G]. Thus the ring A[G] is isomorphic to F [xi,j ][det−1 ]/I where I is the ideal of functions vanishing on G. In our treatment only 4 algebraic subgroups of GL(m, F ) will appear. 2 In

this notes by algebraic group we always mean a linear one.

2. PRELIMINARIES

13

(1) The subgroup Dm of diagonal matrices with entries x1 , . . . , xm . This has as coordinate ring the ring of polynomials in the xi with the inverse of the determinant hence the ring of Laurent polynomials  ±1 A[Dm ] = F [x1 , . . . , xm ][( xi )−1 ] = F [x±1 1 , . . . , xm ]. i

(2) The subgroup Um of strictly upper triangular matrices with entries 1 on the diagonal, 0 under the diagonal, and xi,j , 1 ≤ i < j ≤ m. This has as coordinate ring the ring of polynomials in the xi,j A[Um ] = F [xi,j ], 1 ≤ i < j ≤ m. (3) The subgroup Bm of upper triangular matrices with entries xi = 0 on the diagonal, 0 under the diagonal, and xi,j , 1 ≤ i < j ≤ m. This has as coordinate ring the ring ±1 A[Bm ] = F [x±1 1 , . . . , xm ; xi,j ], 1 ≤ i < j ≤ m.

(4) The subgroup SL(m, F ) of m × m matrices of determinant 1. This has as coordinate ring F [xi,j ]/(det −1). Definition 2.13. A finite dimensional rational representation of a linear algebraic group G ⊂ GL(m, F ) is a homomorphism ρ : G → GL(N, F ) such that the matrix elements ρ(g)h,k , h, k = 1, . . . , N belong to A[G]. For representations W, W1 , W2 of a group G one has the usual notions of dual, direct sum and tensor product of representations (φ ∈ W ∗ , v ∈ W, wi ∈ Wi ): (14) gφ | v := φ | g −1 v,

g(w1 ⊕ w2 ) = gw1 ⊕ gw2 ,

g(w1 ⊗ w2 ) = gw1 ⊗ gw2

One easily verifies Proposition 2.14. Given rational representations W, W1 , W2 of an algebraic group G then W ∗ , W1 ⊕ W2 and W1 ⊗ W2 are rational. In particular a finite dimensional rational representation of GL(m, F ) is a homomorphism ρ : GL(m, F ) → GL(N, F ) such that the matrix elements ρ(g)h,k belong to the algebra F [xi,j ][det−1 ] for all 1 ≤ h, k ≤ N . For GL(m, F ) we also have the notion of polynomial representation. This is a representation ρ : GL(m, F ) → GL(N, F ) such that the matrix elements ρ(g)h,k belong to F [xi,j ]. It is better to take a basis free point of view, so if V is a finite dimensional vector space, isomorphic to F m , we take GL(V )  GL(m, F ) and observe that a polynomial representation of GL(V ) is any homomorphism ρ : GL(V ) → GL(W ) which extends to a multiplicative End(W ) valued polynomial map on End(V ). Typical polynomial representations are the tensor representation mapping ξ ∈ n, and related representations such End(V ), to ξ ⊗n ∈ End(V ⊗n ) which is of degree n as for instance, the exterior power ξ → ξ,n ≤ m = dim V . In particular the m ξ = det(ξ). one dimensional degree m representation ξ → Exercise 2.15. We leave to the reader to verify that every rational representation W of GL(V ) can be written as the tensor product W = V ⊗ deta where V is a polynomial representation and a ∈ Z.

14

INTRODUCTION AND PRELIMINARIES

Given any polynomial representation of GL(V ), which we extend to a polynomial map ρ : End(V ) → End(W ), one has that the function ξ → χρ (ξ) := tr(ρ(ξ)), that is called the character of the representation, is an invariant under conjugation. One of the facts that we shall see is that all invariants are obtained as linear combinations of these basic invariants. Next observe that the coordinate ring A[G] is a representation of G × G by setting: (a, b)f (x) := f (a−1 xb), (a, b) ∈ G × G. The two actions of G are called the left and right action and denoted by g f, f g respectively. Although A[G] is usually infinite dimensional it can also be treated as rational representation in the sense that it is a sum of finite dimensional rational representations. This can be easily proved, since G is algebraic the multiplication map G × G → G induces a map on functions which, to a function f , associates the function f (xy) ∈ A[G × G] = A[G] ⊗ A[G], so f (xy) =

k 

fi (x)gi (y).

i=1

It follows, for instance, that the functions f (xg), g ∈ G, obtained from f (x) under right action of G, span a finite dimensional representation Vf contained in the span of the functions fi (x). If we choose the fi (x) to be a basis of Vf the same formula shows that the representation is rational. The same argument for the left action and also for the action of G × G. A first important example is the coordinate ring of Dm . The group Dm is a torus that is an algebraic group T := {(t1 , . . . , tm ), ti ∈ F ∗ } isomorphic to a product of m copies of the multiplicative group F ∗ . Its coordinate ring has as m basis the monomials i=1 xhi i , hi ∈ Z. Each monomial is the basis element of a m h i 1-dimensional space F i=1 xi which is stable under right and left action. m Given t := (t1 , . . . , tm ) we have that the function f (x) = i=1 xhi i transforms (by the right action) as m m m    f (xt) = (xi ti )hi = xhi i thi i . i=1

i=1

i=1

Therefore Proposition 2.16. The coordinate ring A[Dm ] decomposes as direct sum of hi the irreducible 1-dimensional representations (t1 , . . . , tm ) → m i=1 ti . These are m called characters and are indexed by Z . The coordinate ring A[G] has the important property Theorem 2.17. Every finite dimensional rational representation W of G can be embedded, as sub-representation, in a direct sum A[G]k . Proof. Let f 1 , . . . , f k be a basis of W ∗ . For each v ∈ W we have by definition that f i | xv ∈ A[G]. The map v → f i | xv is G equivariant where we consider on A[G] the right action. Finally the map i : V → A[G]k , i(v) := (f 1 | xv, . . . , f k | xv)

2. PRELIMINARIES

15

is injective since, if v = 0, at least one of the coordinates f i | v is different from 0.  Corollary 2.18. Every rational representation of a torus T is a direct sum of 1-dimensional characters. Proof. By Theorem 2.17 and Proposition 2.16.



The characters appearing in the decomposition of a rational representation V of T are called the weights of V . In particular we may apply this to the torus of diagonal matrices acting on a rational representation of GL(m, F ) and to its center F ∗ , a 1-dimensional torus. We deduce: Proposition 2.19. (1) A rational representation W of GL(m, F ) can be decomposed as W = ⊕d∈Z Wd with Wd = {f ∈ W | ρ(a)w = ad w, ∀a ∈ F ∗ }. (2) If W is polynomial Wd = 0 for each d < 0. If W = Wd for some d we shall say that W is homogeneous of degree d. (3) The tensor product of rational representations is a rational representation and if each has a degree, then the tensor product has as degree the sum of the degrees. Remark 2.20. Remark that a rational representation W of GL(m, F ), under the group Dm of diagonal matrices, has a basis of weight vectors ui such that t · ui = χi (t)ui , ∀t ∈ Dm . From Formula (14) we see that the dual basis ui of W ∗ is a basis of weight vectors of weights χ−1 i . 2.4. A formal approach. We will want to state some of our theorems for modules over the integers or a finite field. In order to do this it is better to develop a more formal approach. The notion of algebraic group G is strictly connected with that of Hopf algebra and of affine group scheme. Take as example the linear group GL(m). For every commutative ring S one may consider the group GL(m, S) of invertible m × m matrices with entries in S. Moreover if f : S1 → S2 is a homomorphism one has an induced homomorphism of groups f : GL(m, S1 ) → GL(m, S2 ), we thus have a covariant group valued functor from the category of commutative rings to groups. Now a covariant set valued functor F(X) on a category C is said to be representable, if there is an object A ∈ obj(C) such that the functor X → F(X) is naturally isomorphic to hA (−) := hom(A, −). Clearly the linear group GL(m) is represented by the commutative algebra Z[xi,j ][d−1 ] where d = det(xi,j ). The following Yoneda’s Lemma is fundamental. We start from an arbitrary covariant functor F : C → Sets with values in the category of sets and then a representable functor hA (−) := hom C (A, −). The set N at(hA , F) of natural transformations between these functors is then given by: Lemma 2.21 (Yoneda). We have a canonical isomorphism between (15)

∼ =

φ : N at(hA , F) −→ F(A),

16

INTRODUCTION AND PRELIMINARIES

given by ϕ → ϕA (1A ). The very definition of tensor product of algebras over some commutative ring A implies that, given two set valued functors from the category of commutative A algebras, represented by two commutative algebras S1 , S2 their product is represented by the algebra S1 ⊗A S2 . Definition 2.22. An affine scheme resp. affine group scheme G defined over some commutative ring A is a representable set (resp. group) valued functor from the category of commutative A algebras. If S represents the affine scheme G, the identity map 1S of S corresponds to the generic element g ∈ G(S). If S is the algebra representing G then by the previous remarks S ⊗A S represents G × G. Then, by Yoneda’s Lemma the multiplication m : G × G → G is encoded into a coproduct Δ : S → S ⊗A S. In order to complete the group axioms one has that the inverse map G → G, g → g −1 is encoded by the antipode I : S → S and finally the unit element 1 ∈ G by the counit η : S → A. Of course all the group axioms translate, by Yoneda’s Lemma, into corresponding axioms for S, Δ, I, η which are usually expressed by various commutative diagrams (cf. [67]). Such a 4-tuple S, Δ, I, η is called a commutative Hopf algebra and Yoneda’s Lemma tells us that it is equivalent to give an affine group scheme or a commutative Hopf algebra. For instance in our case, GL(m) represented by Z[xi,j ][d−1 ], we have that the matrix ξ with entries (xi,j ) is the generic element m  0, i = j −1 xi,h ⊗ xh,j , I(xi,j ) = yi,j , (yi,j ) = ξ , η(xi,j ) = Δ(xi,j ) = 1, i = j h=1 which can be compactly written on the generic element (16)

Δ(ξ) = (ξ ⊗ 1)(1 ⊗ ξ), I(ξ) = ξ −1 , η(ξ) = 1.

One further has Δ(d) = d ⊗ d. The notion of a rational representation correspond to the notion of comodule M for the Hopf algebra S given by a map Δ : M → S ⊗A M satisfying suitable axioms. In particular the usual axiom 1 · m = m becomes η ⊗ 1 ◦ Δ : M → S ⊗A M → M is the identity implying that Δ is injective. In a similar way an action of an affine algebraic group G on an affine variety V is replaced by an action of a group valued functor G on a set valued functor V . If both are representable, respectively by a commutative Hopf algebra S and a commutative algebra B, again Yoneda’s Lemma encodes this into a coproduct Δ : B → S ⊗A B satisfying suitable axioms expressing the fact that B is a comodule over S but also compatible with the algebra structure (this encodes the fact that for all C the group G(C) acts by automorphisms on the coordinate algebra B ⊗A C of V (C)). Then, given any commutative algebra C, we have the usual action of the group G(C) on V (C). For elements g : S → C, x : B → C of g ∈ G(C) and x ∈ V (C) one has g · x : B → C given by (17)

Δ

g⊗x

μ

g · x : B −→ S ⊗A B −→ C ⊗A C −→ C

2. PRELIMINARIES

17

where the last map μ : C ⊗ C → C is multiplication. As an example, consider the natural representation V of GL(V ). The variety V is given by the functor C → C m , C a commutative ring, and is represented by B = Z[x1 , . . . , xm ]. Its generic element is the column vector v of entries xi , Thus the coproduct Δ : B → S ⊗ B is given by m  Δ(xj ) = xi,h ⊗ xh h=1

which on the generic pair (X, v) is equivalent to the usual multiplication map (X, v) → Xv. For every algebra C the algebra B induces an algebra of regular functions on the set V (C) by the tautological map b(x) := x(b), x : B → C. In particular one may define the invariants in B as those elements which, for every algebra C, induce a function on V (C) invariant under G(C). In particular one may think of the coproduct as the action of the generic element of the group G and again by Yoneda’s Lemma one can see that the invariants are the elements invariant under the generic element, that is: (18)

B G := {x ∈ B | Δ(x) = 1 ⊗ x}.

The connection with usual invariants is when one has a density property, that is if an algebra C is such that the map of S ⊗A B to the functions on G(C) × V (C) is injective one has B G(C) = B G . In fact in this case one has that b ∈ B G(C) means that Δ(b) and 1 ⊗ b induce the same function on G(C) × V (C) hence they are equal.

Part I

The classical theory

20

I. THE CLASSICAL THEORY

3. Representation theory 3.1. The representation theory of GL(m) in characteristic 0. The representation theory of GL(m) in characteristic 0 is strictly connected to that of the symmetric group and it is the outcome of the work of Frobenius, Schur, Young among others. See for instance [72], [48] and [41]. We review without proofs the main facts. Theorem 3.1. (1) In characteristic 0 each rational representation is defined over Q. (2) Every rational representation decomposes into a direct sum of irreducible rational representations. (3) The irreducible rational representations of GL(m) are classified by sequences: λ := λ1 ≥ λ2 ≥ . . . ≥ λm ,

with

λi ∈ Z.

m

(4) The sequence 1 , λi = 1, i = 1, . . . , m corresponds to the 1-dimensional representation det given by the determinant.  An irreducible representation has degree d = ni=1 λi and, it is polynomial if and only if λm ≥ 0. We denote by ∇λ (V ) the irreducible representation, defined up to isomorphism,  corresponding to the sequence λ (e.g. ∇1k (V ) = k (V ), ∇k (V ) = S k (V ) ). Given λ := λ1 ≥ λ2 ≥ . . . ≥ λm and i ∈ Z the sequence λ + im = i + λ1 ≥ i + λ2 ≥ . . . ≥ i + λm corresponds to ∇λ (V ) ⊗ deti . When λ := λ1 ≥ λ2 ≥ . . . ≥ λm ≥ 0, so that ∇λ (V ) is a polynomial representation,  we may consider λ as a partition (a Young diagram cf. 8.3) λ  d of degree d= m i=1 λi . In this case we even have a functor on vector spaces, V → ∇λ (V ) and, for maps, f : V → W =⇒ ∇λ (f ) : ∇λ (V ) → ∇λ (W ). Definition 3.2. If λ is a partition, its height, denoted by ht(λ), is the maximum k such that λk = 0. As functor we have ∇λ (V ) = 0 if and only if ht(λ) ≤ dim V . 3.1.1. The Schur functions. The character of a representation of GL(m) is a function which is conjugation invariant and, up to multiplication by some power of det, a function on Mm (F ). By Theorem 1.4 it can be identified with its restriction to diagonal matrices, a symmetric function in the eigenvalues. If λ is a partition the character of ∇λ (V ) is called the Schur symmetric function sλ (x1 , . . . , xm ) (cf. [41] Formula 3.1 p. 40). i Proposition 3.3. (1) We have ∇1i (V ) = V and s1i (x1 , . . . , xm ) = σi is the ith elementary symmetric function. (2) The functions sλ (x1 , . . . , xm ), ht(λ) ≤ m are a basis over Z of the ring of symmetric functions Z[σ1 , . . . , σm ] in m-variables and can be is expressed as universal polynomials in x1 , . . . , xm . (3) sλ (x1 , . . . , xm−1 , 0) = sλ (x1 , . . . , xm−1 ) if ht(λ) < m. Otherwise it equals 0.

3. REPRESENTATION THEORY

21

(4) The abstract ring of symmetric functions Z[σ1 , . . . , σi , . . .], i = 1, . . . , ∞, has as basis the sλ . The kernel of the restriction to Z[σ1 , . . . , σm ] given by setting σj = 0, ∀j > m has as basis the sλ with ht(λ) > m. ±1 ]. It has as basis (5) The character ring of GL(m) equals Z[σ1 , . . . , σm−1 , σm i the elements sλ (x1 , . . . , xm )σm , ht(λ) < m, i ∈ Z. 3.1.2. Tensor symmetry. Let us describe two instances where polynomial representations of GL(V ) appear. I) Let us act on V ⊗N with the two commuting groups, GL(V ) and the symmetric group SN by, for (g, σ) ∈ GL(V ) × SN , (19)

(g, σ)(v1 ⊗ . . . ⊗ vN ) = gvσ−1 (1) ⊗ . . . ⊗ gvσ−1 (N ) . Theorem 3.4 (Schur–Weyl duality). (1) The subalgebras of End(V ⊗N ) generated by GL(V ) and SN are each the centralizer of the other. (2) The kernel of the homomorphism of the group algebra F [SN ] to the algebra . In this case EndGL(V ) (V ⊗N ) is nonzero if and only if N > m = dimF V it is the ideal of F [SN ] generated by the antisymmetrizer σ∈Sm+1 σ σ, σ denoting the sign of the permutation σ. (3) We have the decomposition in isotypic components: V ⊗N = ⊕λN Mλ ⊗ ∇λ (V ),

ht(λ) ≤ dim V

with Mλ irreducible representations of SN . We shall later see how to develop the theory in a characteristic free setting. It will follow that items 1,2 are still valid while item 3 has to be replaced by a canonical filtration. II) Cauchy’s Formula. Given two vector spaces V, W with dim(V ) = m, dim(W ) = n consider S[W ⊗ V ] as representation of the group GL(W ) × GL(V ). Clearly the character of S[W ⊗ V ] as a GL(W ) × GL(V )-module equals n,m 

1 . 1 − xi yj i=1,j=1 Furthermore one has the identity (due to Cauchy) (20)

n,m 

 1 = sλ (y1 , . . . , yn )sλ (x1 , . . . , xm ). 1 − xi yj i=1,j=1 λ

Theorem 3.5 (Representation theory—Cauchy’s formula). In characteristic 0 the Cauchy identity (20) is equivalent to the decomposition: S[W ⊗ V ]  ⊕λ ∇λ (W ) ⊗ ∇λ (V ). In particular • S[End(V )∗ ] = S[V ⊗ V ∗ ] = ⊕λ ∇λ (V ) ⊗ ∇λ (V ∗ ) implies that S[End(V )∗ ]GL(V ) = ⊕λ [∇λ (V ) ⊗ ∇λ (V ∗ )]GL(V ) . • [∇λ (V ) ⊗ ∇λ (V ∗ )]GL(V ) is 1-dimensional, spanned by the character of ∇λ (V ). • Under the identification of S[End(V )∗ ]GL(V ) with the algebra of symmetric functions F [x1 , . . . , xm ]Sm the character of ∇λ (V ) corresponds to the basis of Schur functions.

22

I. THE CLASSICAL THEORY

Remark 3.6. In fact the connection with Schur–Weyl duality comes from the remark that V ⊗N ⊂ S N (F N ⊗ V ) = ⊕λN ∇λ (F N ) ⊗ ∇λ (V ) is the subspace of multilinear elements, that is the elements which transform under the diagonal group N of GL(N, F ) by the character i=1 ti . The decomposition V ⊗N = ⊕λN Mλ ⊗ ∇λ (V ) arises from the fact that Mλ is the space of multilinear elements in ∇λ (F N ), stable under the action of the symmetric group SN ⊂ GL(N, F ). Finally one can show that, for each t, the two algebras of operators induced on S t [W ⊗ V ] by the actions of the two groups GL(W ) and GL(V ) are each the centralizer of the other. 3.2. Invariants of a single matrix. For a single matrix we have Theorem 1.4 establishing an isomorphism between the ring of invariants of a matrix and the ring of symmetric functions. Recall that for a matrix A ∈ End(V ) we have σi (A) = tr(∧i A) where ∧i A is the linear transformation induced by A in ∧i V . When we work over Q there is a combinatorial formula for the polynomial expression of σi (A) in term of elements tr(Aj ). For this we can use the restriction to the diagonal matrices and when A is diagonal, σi (A) is the i-th elementary symmetric function computed in the (diagonal) entries of A while tr(Aj ) is the sum of j-powers of these entries. So it suffices to express the elementary symmetric polynomial σi = σi (x1 , . . . , xm ) in terms of the powers sums ψi = ψi (x1 , . . . , xm ) =  j j+1 y xi1 + . . . + xim . Using the Taylor expansion for log(1 + y) = ∞ j=1 (−1) j , we get m 

σi (x1 , . . . , xm )ti =

m 

(1 + xr t) = exp(

r=1

i=0

∞ 

(−1)j+1

j=1

Expanding and setting σh = 0 for r > m, we deduce r   ((−1)j+1 ψj )hj σr = = (−1)r hj h !j j j=1 h1 +2h2 +···rhr =r h1 ≥0,...,hr ≥0

ψj (x1 , . . . , xm ) j t ). j



h1 +2h2 +···rhr =r h1 ≥0,...,hr ≥0

r  (−ψj )hj . hj !j hj j=1

Thus, Proposition 3.7. Given A ∈ Mm (F ), one has r   ((−1)j+1 tr(Aj ))hj (21) σr (A) = . hj !j hj j=1 h1 +2h2 +···rhr =r h1 ≥0,...,hr ≥0

Remark 3.8. Observe that the previous Formula (21) follows also from Corollary 4.8. In fact by the theory of polarization and specialization σr (x) = Tr (x,...,x) . Then r! one collects in the formula Tr (x, . . . , x) = σ∈Sr Tσ (x) the terms relative to the same cycle structure. Using the fact that the conjugacy class of a permutation with hi cycles of length i has cardinality r r!h !j hj one has the required identity. j=1

j

Let A be an m × m matrix over a commutative Q-algebra. Corollary 3.9. If tr(Ai ) = 0, i = 1, . . . , m, we have Am = 0. If the entries of A are in a ring without nilpotent elements, the converse is also true.

3. REPRESENTATION THEORY

23

Proof. The Cayley–Hamilton theorem states that A satisfies its characteristic polynomial. Hence if tr(Ai ) = 0, i = 1, . . . , m we have that this polynomial reduces to tm and so Am = 0. We leave the last statement as an exercise.  The list of the first expressions of σi (A), i = 1, 2, 3, 4 in term of the elements tr(Ai ) is: σ1 (A)=tr(A),

1 σ2 (A)= (tr(A)2 − tr(A2 )), 2

1 σ3 (A)= (tr(A)3 − 3tr(A)tr(A2) + 2tr(A3 )), 6 σ4 (A) =

1 (tr(A)4 − 6tr(A)2 tr(A2 ) + 3tr(A2 )2 + 8tr(A)tr(A3) − 6tr(A4 )). 24

3.3. FFT for matrices in characteristic 0. The theorem in characteristic 0 is based on a general method, polarization and specialization which can be developed over any infinite field (or even abstractly in the theory of polynomial laws cf. §13) but it works particularly well in characteristic 0 (Arhonold’s method). 3.3.1. Polarization and multilinearization. Given a polynomial function f (v) on a vector space V over an infinite field F , and auxiliary indeterminates t1 , . . . , td ,  we compute f ( di=1 ti vi ) which is a polynomial both in the variables ti and in the vector variables vi ∈ V . We can expand this polynomial in the variables ti so that its coefficients are polynomials in the variables vj and set (22)

f(

d 

tj vj ) :=

j=1



th1 1 . . . thd d fh1 ,...,hd (v1 , . . . , vd )

Taylor expansion.

h1 ,...,hd

The first remark we should make is the symmetry of the operation, if σ is a permu  tation of 1, 2, . . . , d we have f ( di=1 ti vσ(i) ) = f ( di=1 tσ−1 (i) vi ), so that (23)

fh1 ,...,hd (vσ(1) , . . . , vσ(d) ) = fhσ−1 (1) ,...,hσ−1 (d) (v1 , . . . , vd ).

Lemma 3.10. The polynomial fh1 ,...,hd (v1 , . . . , vd ) is homogeneous of degree hi in each variable vi . Proof. We introduce further variables λi and see that  h1 ,...,hd

=

th1 1 . . . thd d fh1 ,...,hd (λ1 v1 , . . . , λd vd ) = f ( 

d 

tj λj vj )

j=1

(λ1 t1 )h1 . . . (λd td )hd fh1 ,...,hd (v1 , . . . , vd )

h1 ,...,hd

=



th1 1 . . . thd d λh1 1 . . . λhd d fh1 ,...,hd (v1 , . . . , vd ).

h1 ,...,hd

Hence fh1 ,...,hd (λ1 v1 , . . . , λd vd ) = λh1 1 . . . λhd d fh1 ,...,hd (v1 , . . . , vd ) as desired.   i Remark 3.11. If we just consider one variable t and write f (tv) = i t fi (v) we have that the polynomials fi (v) are the homogeneous componentsof f (v). If f is homogeneous of degree d we see that the total degree hj of each summand fh1 ,...,hd (v1 , . . . , vd ) is also equal to d.

24

I. THE CLASSICAL THEORY

We are going to use a compact symbolic expression for Formula (22) which, for a homogeneous polynomial of degree d is f(

d 

tj vj ) =



th fh (v1 , . . . , vd ).

h | |h|=d

j=1

 Thus by h we mean some vector h = (h1 , . . . , hd ), hi ∈ N, by |h| = i hi and finally th := th1 1 . . . thd d . Of particular interest is the full polarization or multilinearization that is the term f1,...,1 (v1 , . . . , vd ) which is multilinear and symmetric, by Formula (23), in d vector variables.  3.3.2. Specialization. When in f ( dj=1 tj vj ) we set all the variables vi equal to v, we get the operation of specialization. Assume f homogeneous of degree d, we have: f(

d  j=1

tj v) = (

d  j=1

tj )d f (v) =

 h1 ,...,hd |



th1 1 . . . thd d fh1 ,...,hd (v, . . . , v). hj =d

One thus sees equating coefficients on the two sides, that 

d f (v) = fh1 ,...,hd (v, . . . , v). h1 , . . . , hd In particular when we specialize the full polarization we obtain (24)

d!f (v) = f1,...,1 (v, . . . , v).

For this reason the process of specialization is sometimes called restitution. Notice also that this formula points out to the fact that this formalism works better in characteristic zero than in positive characteristics, a fact which is the source of most difficulties in this last case. Polarizations are clearly linear operators, in fact they are differential operators, a fact that will not play a role in our treatment. This procedure can be applied also for elements of the free algebra, in fact the formalism of tensor algebra is just a modern reformulation of the classical formalism of polarization, cf. §13. Polarization can be iterated, if we have a polynomial f (v1 , v2 , . . . , vm ) in several vector variables we can polarize separately each variable. 3.3.3. Arhonold’s method. In characteristic 0 we may use the classical Arhonold’s method. We take some vector space V and consider polynomial functions in countably many copies of V . By this we mean the union of all algebras of polynomials in any finite number of copies of V . On this space one has the operators of polarization and specialization and one has also the notion of multilinear polynomials. An immediate consequence of Formula (24) is: Theorem 3.12 (Arhonold method). Let A and B be two spaces of polynomials in countably many copies of V both closed under polarization and specialization. If the space of multilinear polynomials in A contains the space of multilinear polynomials in B then A contains B.

3. REPRESENTATION THEORY

25

In fact this formalism can be vastly generalized to any polynomial representation U of some degree n of GL(m, F ), n ≤ m. This has a basis of weight vectors for the diagonal group and we may speak of n-multilinear elements that is those elements of U such that n  ti u. diag(t1 , . . . , tm )u = i=1

We then have as easy consequence of Remark 3.6: Theorem 3.13 (Arhonold method). Under the previous assumptions on U , if A and B are two GL(m, F ) submodules of U and if the n-multilinear elements in A contain the n-multilinear elements in B then A contains B. 3.4. Matrix invariants. A classical application of Arhonold’s method is, given a representation V of a group G, to compute the ring A of invariant polynomial functions on countably many copies of V . It is clear that A is closed under polarization and specialization. One then exhibits some proposed ring of invariants B ⊂ A closed under polarization and specialization. If one can prove that A and B coincide at the level of multilinear elements then they do coincide. Let us illustrate this method in the case of matrix invariants. For a base field F ⊃ Q we want describe the ring of invariants of the action of the general linear group GL(m, F ) acting by simultaneous conjugation on countably many copies of the space Mm (F ) of square matrices. In intrinsic language, we start with an m-dimensional vector space V . Then GL(V ) acts on End(V ) and we want to describe the ring Tm (resp. Tm,n ) of polynomial functions on countably many (resp. n) copies of End(V ) which are invariant under the action of GL(V ). Using polarization and specialization, we analyze Tm by reducing to study multilinear invariants. We fix an alphabet X = (x1 , x2 , . . . , xn ) free variables, generating a free algebra F X = F x1 , x2 , . . . , xn . We denote by (A1 , A2 , . . . , An ) an n-tuple of m × m matrices. We start with the multilinear invariants of n matrices, i.e. the invariant elements of the dual of End(V )⊗n . Lemma 3.14. The multilinear invariants of n matrices are linearly spanned by the functions: (25)

Tσ (A1 , A2 , . . . , An ) := tr(σ −1 ◦ A1 ⊗ A2 ⊗ · · · ⊗ An ), σ ∈ Sn .

If σ = (i1 i2 . . . ih ) . . . (j1 j2 . . . j )(s1 s2 . . . st ) is the cycle decomposition of σ then we have that Tσ (A1 , A2 , . . . , An ) equals (cf. Kostant [35]) (26)

tr(Ai1 Ai2 . . . Aih ) . . . tr(Aj1 Aj2 . . . Aj )tr(As1 As2 . . . Ast ).

Proof. First remark that the dual of End(V )⊗n = End(V ⊗n ) can be identified, in a GL(V ) equivariant way, to End(V ⊗n ) by the pairing formula: A1 ⊗ A2 · · · ⊗ An |B1 ⊗ B2 · · · ⊗ Bn  := tr((A1 ⊗ A2 · · · ⊗ An ) ◦ (B1 ⊗ B2 · · · ⊗ Bn ))  = tr(Ai Bi ).

26

I. THE CLASSICAL THEORY

Under this isomorphism, the multilinear invariants of matrices are identified with the GL(V ) invariants of End(V )⊗n which in turn are spanned, by Theorem 3.4, by the elements of the symmetric group hence by the elements of Formula (25). As for Formula (26), since this identity is multilinear it is enough to prove it on the decomposable tensors of End(V ) = V ⊗ V ∗ which are the endomorphisms of rank 1, u ⊗ φ : v → φ | vu; u ∈ V, φ ∈ V ∗ . So given Ai := ui ⊗ φi and an element σ ∈ Sn in the symmetric group we first verify that σ −1 ◦(u1 ⊗φ1 )⊗(u2 ⊗φ2 )⊗· · ·⊗(un ⊗φn ) = (uσ(1) ⊗φ1 )⊗(uσ(2) ⊗φ2 )⊗· · ·⊗(uσ(n) ⊗φn ) whose trace is the invariant ni=1 φi | uσ(i) . We just use the rules u ⊗ φ ◦ v ⊗ ψ = u ⊗ φ | vψ,

tr(u ⊗ φ) = φ | u.

Ai1 Ai2 . . . Ais = ui1 ⊗φi1 ◦ui2 ⊗φi2 ◦. . .◦uis ⊗φis = φi1 | ui2  . . . φis−1 | uis ui1 ⊗φis .  Theorem 3.15 (FFT for matrices I). The ring Tm of invariants of matrices under simultaneous conjugation is generated by the elements tr(Ai1 Ai2 . . . Aik−1 Aik ), where the formula means that we take all possible non commutative monomials in the Ai and form their traces. Proof. The ring of invariants of matrices contains the ring generated by the traces of monomials and both rings are stable under polarization and specialization. Hence, by Arhonold’s method 3.12, it is enough to prove that they coincide on multilinear elements and this is the content of the previous Lemma.  Invariants of matrices should always be considered together with a more general concept, that of equivariant maps. Definition 3.16. A polynomial map f : Mm (F )k → Mm (F ) is equivariant with respect to the conjugation action of the linear group, if it satisfies (27)

f (gA1 g −1 , . . . , gAk g −1 ) = gf (A1 , . . . , Ak )g −1 , ∀g ∈ GL(m, F ).

For any m, k we denote by Pm,k the space of such maps and call it the space of equivariant maps on k, m×m matrices. Letting k go to infinity, we will also consider Pm the space of equivariant maps on any (finite) number of m × m matrices. Since Mm (F ) is an algebra, and conjugation is a group of algebra automorphisms, the spaces Pm,k and Pm are themselves algebras (non commutative as soon as m, k > 1). Several remarks are in order. First of all, identifying F with the scalar matrices, the ring of invariants functions Tm,k (resp. Tm ) is contained in Pm,k (resp. Pm ). Secondly, the i-th coordinate elements (A1 , . . . , Ak ) → Ai belong to these algebras. Thirdly, if f (A1 , . . . , Ak ) is an equivariant map then tr(f (A1 , . . . , Ak )) ∈ Tm,k is an invariant. There is a simple way to connect equivariant maps with invariants. That is, take f (A1 , . . . , Ak ) an equivariant map and then take a further variable Ak+1 . The function tr (f (A1 , . . . , Ak )Ak+1 ) is an invariant which is linear in Ak+1 . Conversely, starting from an invariant F (A1 , . . . , Ak , Ak+1 ) ∈ Tm,k+1 which is linear in Ak+1

3. REPRESENTATION THEORY

27

and using the fact that the bilinear form tr(AB) is non degenerate one can reconstruct a unique equivariant map f (A1 , . . . , Ak ) such that F (A1 , . . . , Ak , Ak+1 ) = tr (f (A1 , . . . , Ak )Ak+1 ). For the basic invariants Tσ of Formula (26) we take n = k + 1 and we may assume that the last cycle ends with st = k + 1 so the last factor is of the form tr((As1 As2 . . . Ast−1 )Ak+1 ), hence we have that tr(ψσ (A1 , A2 , . . . , Ak )Ak+1 ) := Tσ (A1 , A2 , . . . , Ak+1 ) where ψσ (A1 , A2 , . . . , Ak ) is the equivariant map given by the formula (28)

ψσ (A1 , A2 , . . . , Ak ) = tr(Ai1 Ai2 . . . Aih ) . . . tr(Aj1 Aj2 . . . Aj )As1 . . . Ast−1 .

For example if σ = (1, 5, 3)(4, 2) = (2, 4)(3, 1, 5), ψσ (A1 , · · · , A4 ) = tr(A2 A4 )A3 A1 . We deduce Theorem 3.17 (FFT for matrices II). The ring Pm is generated, over the ring of invariants Tm , by the monomials Ai1 Ai2 . . . Air−1 Air . Remark 3.18. In the FFT we have made no precise estimates on the degree of the monomials necessary to generate. Such estimates exist although there are open problems as to the best possible estimates, we refer to [23] and [48] for a partial discussion. For m = 2 the theory can be made quite precise, cf. [1]. Sometimes it is better to use a more symbolic formulation of the FFT and SFT for matrices. The coordinate ring of Mm (F )n := {(A1 , . . . , An )| Ai ∈ Mm (F )} is the polynomial ring F [xih,k ], h, k = 1, . . . , m; i = 1, . . . , n, xih,k being the function whose value is the (h, k)-coordinate entry of the ith variable matrix Ai . With this language the coordinate map ξi := (A1 , . . . , An ) → Ai can be thought of as the generic matrix ξi = (xih,k ) ∈ Mm (F [xih,k ]). Then the ring Pm is the subring of Mm (F [xih,k ]) generated by the generic matrices ξi and all the traces.

28

I. THE CLASSICAL THEORY

4. Algebras with trace In order to explain the SFT for matrices we develop a language which is a natural evolution of the one developed in the previous paragraphs, at least in characteristic 0, the language of algebras with trace and their identities (cf. [1]). Definition 4.1. An associative algebra with trace, over a commutative ring A is an associative algebra R with a 1-ary operation t:R→R which is assumed to satisfy the following axioms: (1) t is A-linear. (2) t(a)b = b t(a), ∀a, b ∈ R. (3) t(ab) = t(ba), ∀a, b ∈ R. (4) t(t(a)b) = t(a)t(b), ∀a, b ∈ R. This operation is called a formal trace. We denote t(R) := {t(a), a ∈ R} the image of t. Remark 4.2. We have the following implications: Axiom 1) implies that t(R) is an A-submodule. Axiom 2) implies that t(R) is contained in the center of R. Axiom 3) implies that t is 0 on the space of commutators [R, R]. Axiom 4) implies that t(R) is an A-subalgebra and that t is t(R)-linear. We call t(R) the trace algebra of R. The basic example of algebra with trace is of course the algebra of m × m matrices over a commutative ring B with the usual trace. Notice that R does not necessarily have an identity element 1, and if R has a 1 we have made no special requirements on the value of t(1). Of course in many important examples this is a positive integer. Algebras with trace form a category (in fact we should distinguish two categories, since we also have the category of algebras with trace and with 1), where objects are algebras with trace and morphisms algebra homomorphisms φ which commute with the trace t(φ(a)) = φ(t(a)). Recall that in an algebra R with some extra operations, an ideal I must be stable under these operations so that R/I can inherit the structure. In an algebra with trace we assume that I is stable under the trace and call it a trace ideal I. That is if a ∈ I then t(a) ∈ I. The usual homomorphism theorems are then valid for algebras with trace. Given an algebra with trace R, if we forget the trace we just have an associative algebra. On the other hand, there is a simple formal construction which associates to any A algebra R an algebra with trace Rt . Rt := R ⊗ S[R/[R, R]], and given an element r ∈ R, its trace t(r) is the class of r in R/[R, R] (this is a functor adjoint to the previous forgetful functor). In particular for a given a finite or countable alphabet X, one gets a free algebra with trace applying the previous construction to the free algebra AX. In this case the space AX/[AX, AX] has as basis the monomials up to cyclic equivalence. Denoting by t(M ) the cyclic equivalence class of a monomial we

4. ALGEBRAS WITH TRACE

29

have that S[AX/[AX, AX]] is the polynomial ring A[t(M ))] while AXt is the polynomial algebra AX[t(M )] where M runs over all monomials in X taken up to cyclic equivalence. Let us respectively denote by TX := A[t(M ))] and PX := T Xt , the commutative trace algebra and the free algebra with trace in the set of variables X. In particular if X = {x1 , x2 , . . . , xr , . . .} is infinite, one may easily verify: Proposition 4.3. For every k ∈ N the map Ψ : σ → ψσ (x1 , x2 , . . . , xk ) given by the symbolic analogue of Formula (28) is a linear isomorphism of A[Sk+1 ] with the multilinear elements in x1 , . . . , xk , of PX . The map T : σ → Tσ (x1 , x2 , . . . , xk ) given by the symbolic analogue of Formula (25) is a linear isomorphism of A[Sk ] with the multilinear elements in x1 , . . . , xk , of TX . By definition of free algebra, given any A-algebra with trace R the (trace preserving) homomorphisms from PX = AX[t(M )] to R are given by evaluations of polynomials associated to maps X → R. Using evaluations allows us to introduce the notion of trace identity for a trace algebra R, that is a element f ∈ PX = AX[t(M )] such that f and all its multihomogeneous components vanish under all evaluations in R. The condition regarding the multihomogeneous components is automatic if A is an infinite field but we add it since we are really interested only in identities of an algebra R which remain identities under extensions of the coefficient ring A ⊂ B. Let us now remark that the set of all trace identities of a trace algebra R is a special ideal of PX , called a T -ideal, that is an ideal in PX closed under trace and under all substitutions of variables. The theory is really effective only when A is a field of characteristic 0, so from now on, in this section, we assume that this is the case. Let us now fix m and let X be countable. We have a canonical homomorphism π : PX → Pm , π : xi → ξi mapping, for each i, the variable xi to the i-th coordinate function, i.e. the generic matrix ξi . As a consequence of Theorem 3.17 we obtain Corollary 4.4. π is surjective and its kernel is the T -ideal of trace identities for m × m matrices. 4.1. Trace identities. We now use Theorem 3.4.2, to understand trace identities of matrices. For this we will again use the language of universal algebra. We work over a field F of characteristic 0. We have the canonical map compatible with trace π : PX → Pm whose kernel is the ideal of trace identities of matrices. In order to describe this ideal we follow the strategy of exhibiting special identities, construct the T -ideal generated by these identities and then prove that the proposed ideal coincides with the T -ideal of all identities by applying Theorem 3.13, that is we show that the two ideals coincide on multilinear elements. By Proposition 4.3 the map Ψ of Formula (28) is a linear isomorphism of F [Sk+1 ] with the space of multilinear trace polynomials in k variables. From Theorem 3.4 we know that: Theorem 4.5. The map π ◦ T : F [Sk+1 ] → TX → Tm ,    π◦T : aσ σ → aσ Tσ (x1 , x2 , . . . , xk+1 ) → aσ Tσ (ξ1 , ξ2 , . . . , ξk+1 ) σ∈Sk+1

σ∈Sk+1

σ∈Sk+1

30

I. THE CLASSICAL THEORY

has as image the multilinear trace invariants for k + 1 m × m matrices, as kernel the ideal generated by the antisymmetrizer in m + 1 elements (when k ≥ m). The map π ◦ Ψ : F [Sk+1 ] → PX → Pm    π◦Ψ: aσ σ → aσ ψσ (x1 , x2 , . . . , xk ) → aσ ψσ (ξ1 , ξ2 , . . . , ξk ) σ∈Sk+1

σ∈Sk+1

σ∈Sk+1

has as image the multilinear trace polynomial maps for k, m×m matrices, as kernel the ideal generated by the antisymmetrizer in m + 1 elements (when k ≥ m). Thus this ideal of F [Sk+1 ] is identified with the space of multilinear trace identities of m × m matrices, in k variables. We deduce that the multilinear trace identities appear only for k ≥ m and for k = m there is a unique identity (up to scalars) associated to the antisymmetrizer in m + 1 elements. The first step consists in identifying the identities Tm+1 (x1 , . . . , xm , xm+1 ), and Am (x1 , . . . , xm ) corresponding to the antisymmetrizer in F [Sm+1 ]:  Tm+1 (x1 , . . . , xm+1 ) := σ Tσ (x1 , x2 , . . . , xm+1 ) σ∈Sm+1

Am (x1 , . . . , xm ) :=



σ ψσ (x1 , x2 , . . . , xm )

σ∈Sm+1

with σ the sign of the permutation and Tm+1 (x1 , x2 , . . . , xm+1 ) = t(Am (x1 , . . . , xn )xm+1 ). 4.1.1. The Cayley–Hamilton identity. Recall that there is a canonical identity, homogeneous of degree m in 1 variable, called the Cayley–Hamilton identity. Consider the characteristic polynomial of m×m matrix A, χA (λ) := det(λ−A), given by Formula (3). By the Cayley–Hamilton theorem χA (A) = 0. interpret this as a trace identity. Remark thus that tr(Ai ) = m We i want to th Newton function in the eigenvalues αj of a matrix A. Hence j=1 αj is the i Formula 21 gives each coefficient of the characteristic polynomial as a well defined polynomial in the elements t(Ai ). These formulas are independent of m so define elements σi (X) in the symbolic trace algebra. Take the element CHm (x) := xm − σ1 (x)xm−1 + · · · + (−1)m σm (x), for each m ∈ N, as a formal element of the free algebra with trace (in 1 variable x) which vanishes when evaluated in m × m matrices. If we fully polarize this element we get a multilinear trace identity CHm (x1 , . . . , xm ) for m × m matrices, whose terms m not  containing traces arise from the polarization of X and are thus of the form τ ∈Sm xτ (1) xτ (2) . . . xτ (m) . By the uniqueness of the identities in degree m we must have that the polarized Cayley–Hamilton is a scalar multiple of the identity corresponding to the antisymmetrizer and to compute the scalar we may look, in the two identities, at the terms not containing a trace. Clearly x1 x2 . . . xm = ψ(1 2...m m+1) and (1 2...m m+1) = (−1)m , and thus we have finally:

4. ALGEBRAS WITH TRACE

Proposition 4.6. CHm (x1 , . . . , xm ) = (−1)m Am (x1 , . . . , xm ) = (−1)m

31



σ ψσ (x1 , x2 , . . . , xm ).

σ∈Sm+1

With the notations of Proposition 4.3 we have (cf. Lew [38]):  CHm (x1 , . . . , xm ) = (−1)m Ψ( σ σ). σ∈Sm+1

Example 4.7. (m = 2; polarize CH2 (X)). 1 CH2 (X) = X 2 − t(X)X + det(X) = X 2 − t(X)X + (t(X)2 − t(X 2 )). 2 A2 = x1 x2 + x2 x1 − t(x1 )x2 − t(x2 )x1 − t(x1 x2 ) + t(x1 )t(x2 ) A2 (x1 , x2 ) = CH2 (x1 + x2 ) − CH2 (x1 ) − CH2 (x2 ). It is interesting to see what is the polarized form of σm (X). The terms corresponding to the determinant correspond exactly to the sum over all the permutations which fix m + 1. Therefore we deduce: Corollary 4.8. The polarized form of σm (X) is the expression  Tm (x1 , . . . , xm ) = σ Tσ (x1 , . . . , xm ). σ∈Sm

Example 4.9. T3 equals t(x1 x2 x3 )+t(x2 x1 x3 )−t(x1 )t(x2 x3 )−t(x2 )t(x1 x3 )−t(x1 x2 )t(x3 )+t(x1 )t(x2 )t(x3 ). Let us look at some implications for relations among traces. Observe that, when we restitute Tm+1 we get t(CHm (x)x) = t(xm+1 ) − t(x)t(xm ) + · · · + (−1)m det(x)t(x) =

Tm+1 (x, x, . . . , x) . (m + 1)!

The vanishing of this expression for m × m matrices is precisely the identity which expresses the m + 1 power sum t(Am+1 ) of m indeterminates as a polynomial in the lower power sums, e.g m = 2: 1 t(A · CH2 (A)) = t(A3 ) − t(A)t(A2 ) + (t(A)2 − t(A2 ))t(A). 2 On 2 × 2 matrices t(A · CH2 (A)) = 0 implies 3 1 t(A)t(A2 ) − t(A)3 2 2 which expresses the fact that, calling a1 , a2 the two eigenvalues of a two by two matrix 3 1 a31 + a32 = (a1 + a2 )(a21 + a22 ) − (a1 + a2 )3 2 2 an identity of symmetric functions. t(A3 ) =

Remark 4.10. From the Cayley–Hamilton identity taking traces we get 0 = t(Am ) − σ1 (A)t(Am−1 ) + · · · + (−1)m m σm (A). This is NOT a trace identity but rather the recursive relation between Newton and elementary symmetric functions.

32

I. THE CLASSICAL THEORY

Before we proceed, we should remark that any map X → TX X defines, by the universal property, a endomorphism by substitution of TX X which sends the trace ring TX into itself. Thus it also makes sense to speak of a T-ideal in TX . In particular we have the T-ideal of trace relations, kernel of the evaluation map of TX into Tm the algebra of invariants of m × m matrices. In order to proceed we need, for n ≥ m: Lemma 4.11. Every permutation γ in Sn+1 can be written as a product γ = α◦β where α ∈ Sm+1 and, in each cycle of β, there is at most 1 element in 1, 2, . . . , m+1. Proof. Observe that, for a, b, c . . . numbers and A, B, C, . . . strings of numbers, so that the followings are permutations, we have: (29)

(a A b B c C . . . e E) = (a b c . . . )(A a)(B b)(C c) . . . (E e).



Example 4.12. If A = 7, 5, 4; B = 6, 3; a = 1, b = 2, (1, 7, 5, 4, 2, 6, 3) = (1, 2)(7, 5, 4, 1)(6, 3, 2). Theorem 4.13 (SFT). i) The T-ideal of trace relations, kernel of the evaluation map of TX into Tm is generated (as a T-ideal) by the trace relation Tm+1 . ii) The T-ideal of (trace) identities of m × m matrices, in the free algebra without 1, is generated (as a T-ideal) by the Cayley–Hamilton identity, while in the free algebra with 1 we have to add the identity t(1) = m. Proof. i) From all the remarks made it is sufficient to prove that a multilinear trace relation (resp. identity) (in the first n variables) is in the T-ideal generated by Tm+1 (resp.Am ). Let us first look at trace relations. By the  description of trace identities it is enough to look at the relations of the type T (τ ( σ∈Sm+1 σ σ)γ), τ, γ ∈ Sn+1 .  We write such a relation as T (τ ( σ∈Sm+1 σ σ))γτ τ −1 ). We have seen that conjugation corresponds to permutation of variables, an operation allowed in the T-ideal, thus we can assume that τ = 1. From Lemma 4.11 we can write γ = αβ with α ∈ Sm+1 and in each cycle of β, there is at most 1 element in 1, 2, . . . , m + 1. Now    σ σ)γ = ( σ σ)αβ = ±( σ σ)β ( σ∈Sm+1

σ∈Sm+1

σ∈Sm+1

so that substituting γ with β, we can assume that γ has the property that in each cycle of γ there is at most 1 element in 1, 2, . . . , m + 1. Since γ has this property, for every σ ∈ Sm+1 the cycle decomposition of the permutation σ ◦ γ is obtained by formally substituting in the cycles of σ, to every element a ∈ [1, . . . , m + 1] the cycle of γ containing a as a word (written formally with a at the end) and then adding all the cycles of γ not containing elements ≤ m + 1. We interpret this operation in terms of the corresponding trace elements Tσγ (x1 , . . . , xn+1 ) and Tσ (x1 , . . . , xm+1 ). We see that the resulting element Tσγ (x1 , . . . , xn+1 ) is the product of a trace element corresponding to the cycles of γ not containing elements ≤ m + 1 and the element obtained by substituting in Tσ (x1 , . . . , xm+1 ), for the variables xi , the monomials Mi xi = xj1 xj2 . . . xjk xi read off from the cycle (j1 , j2 , . . . , jk , i) containing i in the cycle decomposition of γ, as in Formula (29).

4. ALGEBRAS WITH TRACE

33

As a result we have proved that a trace relation is in the T-ideal generated by Tm+1 . ii) Let us pass now to trace identities. We work with the free algebra without 1. First we remark that, by the definition of ideal in a trace ring, the relation Tm+1 is a consequence of the Cayley–Hamilton identity. A multilinear trace polynomial f (x1 , x2 , . . . , xn ) is a trace identity for matrices if and only if t(f (x1 , x2 , . . . , xn )xn+1 ) is a trace relation. Thus from the previous argument we have that t(f (x1 , x2 , . . . , xn )xn+1 ) is a linear combination of elements of type Z Tm+1 (M1 , . . . , Mm+1 ) = Z t(Am (M1 , . . . , Mm )Mm+1 ), where Z is some trace expression which is multilinear in all the variables not appearing in t(Am (M1 , . . . , Mm )Mm+1 ) and the Mj ’s are monomials in the xi ’s. We have to consider two cases, the variable xn+1 appears in Z or in one of the Mi . In the first case we have Z = t(Bxn+1 ), with B some trace polynomial hence Z Tm+1 (M1 , . . . , Mm+1 ) = t(Tm+1 (M1 , . . . , Mm+1 )Bxn+1 ), and Tm+1 (M1 , . . . , Mm+1 )B is a consequence of the Cayley–Hamilton identity. In the second, due to the antisymmetry of Tm+1 , we can assume that xn+1 appears in Mm+1 = Bxn+1 C. Hence Z t(Am (M1 , . . . , Mm )Mm+1 ) = Z t(Am (M1 , . . . , Mm )Bxn+1 C) = t(CZAm (M1 , . . . , Mm )Bxn+1 ). CZAm (M1 , . . . , Mm )B is also clearly a consequence of the Cayley–Hamilton identity. We leave to the reader to verify that this proof works also for the free algebra with 1 once we normalize t(1) = m.  Remark 4.14. One can prove that for the free   algebra with 12 the m-Cayley– Hamilton identity implies that t(1) is an element m j ej = j=1 jej with ej = ej , k 1, eh ek = δh eh and the free algebra modulo this identity is the direct sum of the algebras Pj , j = 1, . . . , m. 4.2. Infinite matrices. Consider the embedding jm : Mm (A) → Mm+1 (A), with A a commutative ring, as the left upper block in Mm+1 (A). We can then pass to the limit and consider the algebra M∞ (A) = lim Mm (A) →

of infinite matrices with a finite number of non zero entries. Notice that this algebra does not have a 1 so we shall evaluate in M∞ (A) only the free algebra without 1. Given a ∈ Mm (A) the characteristic polynomial of jm (a) ∈ Mm+1 (A) is given by det(λ − jm (a)) = λ det(λ − a). This means equating coefficients, that σi (a) = σi (j(a)), i ≤ m, while σm+1 (j(a)) = 0. We deduce that for each i ≥ 1, we have defined a polynomial function (by this we mean that its restriction to each Mm (A) is polynomial) σi , homogeneous of degree i, on M∞ (A). Given a set X (an alphabet), take the free ring without identity Z+ X over the set X. By the definition of a free ring, we can identify M∞ (A)X with the set of algebra homomorphisms from Z+ X to M∞ (A).

34

I. THE CLASSICAL THEORY

The set A of all polynomial maps M∞ (A)X → M∞ (A) is an algebra by pointwise sum and multiplication. We get a homomorphism J : Z+ X → A by setting J(f )(φ) = φ(f ) for each φ ∈ M∞ (Z)X . From Theorem 4.13, we deduce that, when A = Z, J is injective, therefore Z+ X can be identified to a sub algebra of the algebra A. In particular we may consider an element x ∈ X as the generic infinite matrix or better, a matrix variable, given by the evaluation of a map γ : X → M∞ (A) in x. If we want to consider the elements of X itself as matrix variables we are going to denote this generic infinite matrix simply by x. On the other hand, if we fix an integer m > 0, we are going to denote, as we have already done, the generic m × m matrix given by the evaluation of a map γ : X → Mm (A) in x, by ξx or by ξ if no confusion arises. On the space M∞ (Z) we have defined, for each i ≥ 1, the polynomial functions σi : M∞ (Z) → Z. Thus, if f : M∞ (Z)X → M∞ (Z) is a polynomial map, the composition σi ◦ f : M∞ (Z)X → Z is a polynomial function. σ1 will be often denoted by tr. Composing with the embedding J, we have that σi gives a polynomial map from the free algebra Z+ X to the algebra of polynomial functions on M∞ (Z)X . In particular for any word p in the elements of X, we get a polynomial function σi (p) on M∞ (Z)X . Recall that, by Remark 1.11, we have that σi (p) = σi (p ) if p and p differ by a cyclic permutation.

4.2.1. Amitsur’s formula. We are going to show a remarkable identity due to Amitsur [2], which was later rediscovered and proved in a different way by Reutenauer and Sch¨ utzenberger, [53]. For a matrix variable x consider the two generating series: T [x, t] :=

∞ 

tr(xj )tj−1 , B[x, t] :=

j=1

∞ 

(−1)i σi (x)ti .

i=0

We can then rephrase Proposition 3.7 as the identity  B[x, t] = exp(−

 T [x, t]dt),

T [x, t]dt =

∞  tr(xj ) j=1

j

tj .

 Replace x with i τi xi where the xi are again matrix variables while the τi s are commutative variables. Given a (non commutative) monomial (or a word) w in the xi ’s, we set τ ν(w) to . , hi , . . .) be the evaluation of w in the specialization xi → τi . Thus ν(w) = (h1 , h2 , . . where hi is the number of occurrences of xi in w. Let (w) = |ν(w)| = i hi be the length of the monomial. We have, since trace is linear: (30)

(

 i

τ i xi ) n =

 w | (w)=n

τ ν(w) w =⇒ tr((

 i

τ i xi ) n ) =

 w | (w)=n

τ ν(w) tr(w).

4. ALGEBRAS WITH TRACE

35

Also, using Formula (30), by Lemma 1.8 and recalling that Wp denotes the set of primitive monomials and W0 denotes the set of Lyndon words, cf. Remark 1.9: ∞     T[ τi xi , t] = τ ν(w) tr(w)t(w)−1 = τ iν(w) tr(wi )ti(w)−1 i

w∈Wp i=1

w∈W

=

=



(w)

∞ 

w∈W0

i=1



∞ 

w∈W0

(w) 

=⇒ T [

τ iν(w) tr(wi )ti(w)−1 t(w)(i−1) tr((τ ν(w) w)i )t(w)−1

i=1

τi xi , t]dt =

i



T [τ ν(w) w, t(w) ]d(t(w) ).

w∈W0

We deduce then:    τi xi , t] = exp(− T [τ ν(w) w, t(w) ]d(t(w) )) B[ i

=



 exp(−

w∈W0



T [τ ν(w) w, t(w) ]d(t(w) )) =

w∈W0

B[τ ν(w) w, t(w) ].

w∈W0

 Comparing the terms of degree n in t in B[ i τi xi , t] = w∈W0 B[τ ν(w) w, t(w) ] we deduce Theorem 4.15 (Amitsur [2]). (31)   τ i xi ) = σn ( i

(−1)n−



ji

τ

k i=1

ji ν(pi )

σj1 (p1 ) . . . σjk (pk ).

p1 0, ji (pi )=n

Corollary 4.16. Given any polynomial expression f (x1 , . . . , xk , . . .), (non commutative), in the matrix variables xi , with coefficients in some commutative ring A, we have that each σj (f (x1 , . . . , xk , . . .)) can be expressed as a polynomial in the elements σh (p), p ∈ W0 with coefficients in A.  Proof. A polynomial is a linear combination i αi mi of monomials, we substitute in (31), ti with αi and xi with mi we deduce a polynomial in the elements σi (w), w ∈ W . If the monomial w = pj is non primitive, then σi (pj ) is an invariant polynomial  function of p. It thus a polynomial in the functions σr (p) by Remark 1.5. Example 4.17. For σ2 (a + b), Formula (31) gives σ2 (a + b) = σ2 (a) + σ2 (b) − tr(ab) + tr(a)tr(b) = σ2 (a) + σ2 (b) − σ1 (ab) + σ1 (a)σ1 (b). Of course if the matrices are 2 × 2 this gives a formula for the determinant of their sum. Let us analyze what we get for σ3 (a + b + c). The possible primitive monomials appearing have length at most 3 and the corresponding Lyndon words are, (ordered by degree and lexicographically) a, b, c, ab, ac, bc, a2 b, a2 c, ab2 , abc, acb, ac2 , b2 c, bc2

36

I. THE CLASSICAL THEORY

only elements of degree ≤ 3 may appear. Hence only σi (a), σi (b), σi (c), i ≤ 3; σ1 (ab), σ1 (ac), σ1 (bc), σ1 (a2 b), σ1 (a2 c), σ1 (ab2 ), σ1 (abc), σ1 (acb), σ1 (ac2 ), σ1 (b2 c), σ1 (bc2 ). We then have to write all possible products of degree 3 and sum them with signs, it gives a rather long expression: σ3 (a + b + c) = σ1 (a)σ1 (b)σ1 (c) + σ1 (a)(σ2 (b) + σ2 (c)) + σ1 (b)(σ2 (a) + σ2 (c)) + σ1 (c)(σ2 (a) + σ2 (b)) − (σ1 (a) + σ1 (b) + σ1 (c))(σ1 (ab) + σ1 (ac) + σ1 (bc))

(32)

+ σ3 (a) + σ3 (b) + σ3 (c) + σ1 (abc) + σ1 (acb) + σ1 (a2 b) + σ1 (a2 c) + σ1 (ab2 ) + σ1 (b2 c) + σ1 (ac2 ) + σ1 (bc2 ).

In particular we may group together in Formula (31) all terms which have the km same degree in each variable, that is the coefficient of τ1k1 . . . τm . That is we apply polarization to the coefficient of the characteristic polynomial σn (x).   km σn ( τ i xi ) = τ1k1 . . . τm σk1 ,...,kh (x1 , . . . , xk ). k1 ≥0,...km ≥0,

i



ki =n

From Theorem 4.15 one deduces Theorem 4.18. The functions σh1 ,...,hs (x1 , . . . , xs ) are given by the universal polynomials with integer coefficients in the functions σj (p), where p is a primitive monomial in the variables xi , extracted from Formula (31). We have some obvious symmetries. In the previous example we have σ2,1 (a, b) = −σ1 (a)σ1 (ab) + σ1 (b)σ2 (a),

σ1,1,1 (a, b, c)

= σ1 (a)σ1 (b)σ1 (c) − σ1 (a)σ1 (bc) − σ1 (b)σ1 (ac) − σ1 (c)σ1 (ab) + σ1 (abc) + σ1 (acb) It is worth to point out that σ1,...,1 is just the full polarization of σm (x) (see §3.3). Here is a special case which we shall use in §18.3.4. Lemma 4.19. The polarized form σr,1 (x, y) of σr+1 (x) is (−1)rσ1 (CHr (x)y). Proof. Since this is a formal identity it is enough to prove it over Z and hence even over Q. By Amitsur’s formula (31) we have (33)   (−1)n− ji σj1 (p1 ) . . . σjk (pk ). σr,1 (x, y) = j1 >0....,jk >0,

p1 i. Consider for the opposite algebra Aop , the corresponding ordered list of simple modules E op (i) := D(E(i)) with their injective hulls Qop (i) := D(P (i)) and projective covers D(Q(i)). Correspondingly, we get Δop (i), ∇op (i) and the following proposition is clear Proposition 6.2. Δop (i) = D(∇(i)), ∇op (i) = D(Δ(i)). In view of this Proposition we can prove either one of dual statements regarding the Δ or ∇ modules. We have Lemma 6.3. (1) If h > i, homA (P (h), Δ(i)) = 0, homA (∇(i), Q(h)) = 0. In particular homA (Δ(h), Δ(i)) = 0, and homA (∇(i), ∇(h)) = 0. (2) If the simple module E(k) is a composition factor of Δ(h) (res. ∇(h)), then k ≤ h. (3) Ext1A (Δ(j), Δ(i)) =Ext1A (∇(i), ∇(j)) = 0 for i ≤ j. Proof. 1) Let f : P (h) → Δ(i) be a homomorphism. Since P (h) is projective, f lifts to a homomorphism P (h) → P (i) which then by definition has image in U (i) so maps to 0 in Δ(i). 2) If E(k) is a composition factor of Δ(h), there is a submodule N ⊂ Δ(h) having E(k) as quotient. The morphism P (k) → E(k) lift to a non zero morphism P (k) → N → Δ(h) which implies, by 1), that k ≤ h. 3) Consider, for i ≤ j, an extension 0 → Δ(i) → N → Δ(j) → 0. From our definition we have a quotient map q : P (j) → Δ(j) with kernel U (j). The map q lifts to a homomorphism q : P (j) → N which restricts to a morphism q˜U : U (j) → Δ(i). q

0 −−−−→ U (j) −−−−→ P (j) −−−−→ Δ(j) −−−−→ 0 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ q¯ q˜U  Id 0 −−−−→ Δ(i) −−−−→

N

−−−−→ Δ(j) −−−−→ 0

By definition U (j) is the sum of images of homomorphisms s : P (h) → P (j) with h > j so by the first part, the homomorphism q˜U ◦ s : P (h) → Δ(i) must be 0 since  h > j ≥ i. Thus q˜U = 0 and so q¯ factors through q and our extension splits.

44

II. QUASI-HEREDITARY ALGEBRAS

Let us set Δ(i) ≤ Δ(j), ∇(i) ≤ ∇(j) if i ≤ j. Definition 6.4. We say that a filtration 0 = Mt+1 ⊂ Mt ⊂ · · · ⊂ M1 = M is a Δ-filtration if, for all h, Mh /Mh+1 is one of the modules Δ(i). We say that a filtration 0 = M0 ⊂ M1 ⊂ · · · ⊂ Mt = M is ∇-filtration if, for all h, Mh /Mh−1 is one of the modules ∇(i).

Notice that if a module M has a Δ-filtration, then its dual has a ∇-filtration as a Aop module. Define now F (Δ) (resp. F (∇)) to be the full subcategory of the category of modules formed by those modules M which have a Δ (resp. ∇) filtration.

Corollary 6.5. If a module M has a Δ (resp. ∇), filtration then it has a filtration where for all h, Mh /Mh+1 ≥ Mh−1 /Mh (resp. Mh /Mh−1 ≤ Mh+1 /Mh ). Moreover it has a filtration 0 = Mm+1 ⊂ Mm ⊂ · · · ⊂ M1 = M such that Mh /Mh+1 = Δ(h)nh (resp. a filtration 0 = M0 ⊂ M1 ⊂ · · · ⊂ Mm = M such that Mh /Mh−1 = ∇(h)nh ) for all h (nh ≥ 0)). Proof. If one has a filtration for which our condition is not satisfied we show how one can stepwise modify it in order to satisfy it. For this it is enough to look at the two step filtration 0 −−−−→ Mh /Mh−1 −−−−→ Mh+1 /Mh−1 −−−−→ Mh+1 /Mh −−−−→ 0 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ = = = 0 −−−−→

Δ(i)

−−−−→ Mh+1 /Mh−1 −−−−→

Δ(j)

−−−−→ 0.

By part 2) of lemma 6.3, if i ≤ j this extension splits so we can rearrange the filtration. The second part also follows by the same Lemma.  We give now an alternative description of F(Δ) and F(∇). Notice that Δ(m) := P (m) is projective. For each i we can cut A to eliminate all the P (j) s, j > i, and make Δ(i) projective. To be precise: Let us start with F(Δ). Consider any A module M . For a given j write P (j) = Aej with ej a primitive idempotent. Set M (j) equal to be the sum of all images of morphisms P (j) → M . We claim that  Lemma 6.6. M (j) = Aej M , in particular U (i) = j>i Aej Aei . Proof. In fact since P (j) = Aej is a direct summand of A, every morphism f : P (j) → M extends to a morphism A → M which is just of the form a → ah for an element h ∈ M . Thus the image of f is Aej h. Since h is arbitrary the claim follows.  We can define a decreasing filtration of M M = M(1) ⊃ M(2) ⊃ · · · ⊃ M(m) ⊃ M(m+1) = {0}  by setting M(i) := M/M (j) is naturally an A/Aej A j≥i M (j). In particular  module and M/M(i) is naturally an A/ j≥i Aej A module.

6. GOOD FILTRATIONS AND QUASI-HEREDITARY ALGEBRAS

45

Dually we can define a increasing filtration of M by setting M (i) to be the intersection of allthe kernels of the maps M → Q(j), j ≥ i. In particular M (i) is naturally an A/ j≥i Aej A module. If M = A, A(j) = Aej A which equals the sum of all images of morphisms P (j) = Aej → A, is the two  sided ideal generated by P (j). We then have A(i) = j≥i Aej A and the chain A = A(1) ⊃ A(2) ⊃ · · · ⊃ A(m) ⊃ A(m+1) = {0} is a chain of two sided ideals. For any module M , M(i) = A(i) M . Moreover the images of the ideals Aej A, modulo the radical J, form the various simple matrix blocks of the decomposition. In particular A(i) = A(i−1) , that is ei ∈ / A(i−1) . We now have Proposition 6.7. F(Δ) consists of the modules M such that, for every i, M(i) /M(i+1) = A(i) M/A(i+1) M is projective as a A/A(i+1) -module. F(∇) consists of the modules M such that, for every i, M (i+1) /M (i) is injective as a A/A(i) -module. Proof. Since P (i) = Aei we have A = P (i) ⊕ A(1 − ei ). Any two sided (or even right) ideal D decomposes as D = Dei ⊕ D(1 − ei ) and Dei = D ∩ P (i). Furthermore we have that any map of a module M to P (i) is the composition of a map M → A with the projection A → Aei . So cf. lemma 6.6, A(i+1) ∩ P (i) = U (i) and A(i+1) = U (i) ⊕ (A(i+1) ∩ A(1 − ei )). Thus A/A(i+1) = P (i)/U (i) ⊕ A(1 − ei )/(A(i+1) ∩ A(1 − ei )) and Δ(i) = P (i)/U (i) is a projective A/A(i+1) -module. We may consider Δ(i) as a module over the algebra A(i) /A(i+1) = Aei A/(Aei A ∩ A(i+1) ). In this algebra we have just one simple block modulo the radical and so just one simple module up to isomorphism, the module E(i) whose projective cover is Δ(i). Let now M be an A module and consider the filtration M = M(1) ⊃ M(2) ⊃ · · · ⊃ M(m) ⊃ M(m+1) = {0} The module M(i) /M(i+1) is projective as A/A(i+1) -module if and only if it is projective as A(i) /A(i+1) -module, so if and only if it is isomorphic to Δ(i)hi which implies that M lies in F(Δ). Conversely, if M lies in F(Δ) in view of Corollary 6.5, there is a filtration 0 = M0 ⊂ M1 ⊂ . . . ⊂ Mm = M such that Mm−i+1 /Mm−i  Δ(i)hi . We remark that since, for h > j, hom(P (h), Δ(i)) = 0, A(i+1) M ⊂ Mm−i . We claim that Mm−i is the sum of homomorphisms P (h) → M with h > i. Indeed assume that this is true for Mm−i−1 (for m − i = 0, M0 = {0}). Consider the exact sequence 0 → Mm−i−1 → Mm−i → Δ(i + 1)hi+1 → 0. Take the quotient map q : P (i + 1)hi+1 → Δ(i + 1)hi+1 . This map lifts to a map q˜ : P (i + 1)hi+1 → Mm−i such that Im q˜ + Mm−i−1 = Mm−i proving our claim. It follows that Mm−i ⊂ A(i+1) M . So Mm−i = A(i+1) M for every i and the claim follows. The remaining statement is dual. 

46

II. QUASI-HEREDITARY ALGEBRAS

Proposition 6.8. The categories F(Δ) and F(∇) are closed with respect to taking direct summands. Proof. This follows from Proposition 6.7 and the fact that a direct summand of a projective (rest. injective) module is projective (rest. injective).  Lemma 6.9. If i > j homA (Δ(j), Q(i)) = homA (P (i), ∇(j)) = 0. homA (Δ(i), ∇(j)) = 0 if and only if i = j. Proof. Let us prove the first statement, the proof of the second being completely analogous. Since the socle of Q(i) is E(i), homA (Δ(j), Q(i)) = 0 implies that E(i) appears in a composition series of Δ(j). So we can apply Lemma 6.3.2. Now assume that homA (Δ(i), ∇(j)) = 0. Composing with the injective map ∇(j) → Q(j) we have homA (Δ(i), Q(j)) = 0 and i ≤ j. Composing with the surjection P (i) → Δ(i) we have homA (P (i), ∇(j)) = 0 and thus i ≥ j. Finally if i = j, we have the composition Δ(i) → E(i) → ∇(i).  Definition 6.10. An algebra A is called quasi hereditary with respect to an ordering of its simple modules E(1), . . . , E(m), if (1) As left module A is in F(Δ). (2) EndA (Δ(i)) = F. We point out that the definition depends on the chosen ordering of the simple modules, so it should be more correct to define a quasi hereditary algebra as a pair (algebra, ordering of its simple modules) satisfying our definition. By remark 5.8 E(i) is the top (resp. socle) of Δ(i) (resp. ∇(i)). Furthermore Proposition 6.11. If A is quasi-hereditary E(i) neither appears in a composition series of JΔ(i) nor in a composition series of ∇(i)/E(i). Proof. We have already seen that E(i) is the top of Δ(i). Now, whenever we have an occurrence of E(i) in a composition series we have a submodule N with quotient E(i) and we have a surjective map P (i) → N ⊂ Δ(i). By Lemma 6.3, and the definition of Ui , this map vanishes on Ui so it induces a non zero map of Δ(i) to Δ(i) which, if the algebra is quasi-hereditary is a non zero multiple of the identity. So N cannot be a proper submodule and the claim follows. The other statement is dual.  From now on we are going to assume that A is quasi-hereditary. Lemma 6.12. Assume M is a module in F(Δ). Let h be the maximum such that Δ(h) appears as a sub-quotient of a Δ-filtration of M . Then if f : Δ(h) → M is a non-zero homomorphism, f is injective and cokerf has a Δ-filtration. Assume M is a module in F(∇). Let h be the maximum such that ∇(h) appears as a sub-quotient of a ∇-filtration of M . Then if f : M → ∇(h) is a non-zero homomorphism, f is surjective and ker f has a ∇-filtration. Proof. As usual we prove the first statement. The second follows dually. We can always arrange, by Corollary 6.5 and Lemma 6.3, the filtration of M , as a filtration M = M1 ⊃ M2 ⊃ · · · ⊃ Mr+1 = {0} so that Mr  Δ(h)⊕s while M/Mr has in a composition series only components E(k) with k < h. We claim that every f ∈ hom(Δ(h), M ) factors through the inclusion i : Mr → M . Indeed suppose we had a homomorphism φ : Δ(h) → M which induces a non zero homomorphism φ¯ : Δ(h) → M/Mr . Since E(h) is the top of Δ(h) it is a

6. GOOD FILTRATIONS AND QUASI-HEREDITARY ALGEBRAS

47

composition factor of every nonzero quotient of Δ(h), so im φ¯ and hence also M/Mr have a composition factor isomorphic to E(h), a contradiction. Now take f ∈ hom(Δ(h), M ) and factor f = i ◦ f¯, f¯ : Δ(h) → Δ(h)⊕s = Mr . For at least one of the j = 1, . . . s, pj ◦ f¯ = 0, pj being the projection of Δ(h)⊕s onto its j-th factor. It follows that pi ◦ f¯ is a non zero multiple of the identity so f¯ is into  and cokerf¯  Δ(h)⊕s−1 . This clearly implies that cokerf has a Δ-filtration. We have already seen that, if M is a module in F(Δ), the filtration M = M(1) ⊃ M(2) ⊃ · · · ⊃ M(m) ⊃ M(m+1) = {0} has the property that M(i) /M(i+1)  Δ(i)⊕hi for some hi ≥ 0 and one immediately sees that this is the unique filtration with this property. Dually if M is in F(∇), one has the unique filtration {0} = M (0) ⊂ M (1) ⊂ · · · ⊂ M (m) = M with M (i) /M (i−1)  ∇(i)⊕hi . We are going to call these the canonical Δ (resp. ∇) filtration. h

k

Proposition 6.13. 1) i) Let 0 → N → M → L → 0 be an exact sequence. Assume that M and L are modules in F(Δ), then N is in F(Δ). ii) Furthermore L(i) = k(M(i) ) and N(i) = M(i) ∩ N . h

k

2) i) Let 0 → N → M → L → 0 be an exact sequence. Assume that M and N are modules in F(∇), then L is in F(∇). ii) Furthermore L(i) = k(M (i) ) and N (i) = M (i) ∩ N . Proof. Let as usual prove statement 1). First 1) i). Let us proceed by induction on d = dim M . If d = 0 there is nothing to prove. Let now M = {0} and h be the maximum index for which the module Δ(h) appears in a Δ-filtration of M. Take a non zero f : Δ(h) → M . By Lemma 6.12, f is injective and coker f has a Δ-filtration. Assume that L has a Δ-filtration. Notice that, by Lemma 6.3, the maximum index for which Δ(h) appears in a Δ-filtration of L is at most h since otherwise in a composition series of L and so also necessarily of M would appear a simple component E(k) with k > h. Let us compose f with k getting k◦f = f˜ : Δ(h) → L. If f˜ is non zero then, again by Lemma 6.12, it is injective and we get an exact sequence 0 → N → M/f (Δ(h)) → L/f˜(Δ(h)) → 0. Since both M/f (Δ(h)) and L/f˜(Δ(h)) are in F(Δ) and dim M > dim M/f (Δ(h)), by induction also N lies in F(Δ). Assume now f˜ = 0 so that f (Δ(h)) ⊂ N . We get and exact sequence 0 → N/f (Δ(h)) → M/f (Δ(h)) → L → 0 By induction N/f (Δ(h)) is in F(Δ) and we have the short exact sequence 0 → Δ(h) → N → N/f (Δ(h)) → 0. We deduce that N lies in F(Δ). Let us now prove the statement 1) ii). First of all we recall that M(i) = A(i) M . Applying k we get k(M(i) ) = A(i) k(M ) = A(i) L.

48

II. QUASI-HEREDITARY ALGEBRAS

On the other hand, we clearly have that A(i) N ⊂ A(i) M for each i. We need to show that A(i) N ⊃ A(i) M ∩ N . We have that A(i) M = {0} for i > h so we may assume i ≤ h. As before consider and injection f : Δ(h) → M , and f (Δ(h)) ⊂ A(h) M . Consider the projection π : M → M/f (Δ(h)) = M . We have seen that either the restriction of π to N is injective or f (Δ(h)) ⊂ N . Take n ∈ N ∩ A(i) M . In the first case π injects N into M . Then we identify n = π(n) ∈ N ∩ A(i) M . By induction n ∈ A(i) N . In the second case set N = π(N ) = N/f (Δ(h)) and π(n) ∈ N ∩ A(i) M . So by induction π(n) ∈ A(i) N = π(A(i) N ). This means that there exists a n ∈ A(i) N such that n − n ∈ f (Δ(h)). But f (Δ(h)) ⊂ A(h) N ⊂ A(i) N so n ∈ A(i) N .  h

k

0 → N(i) → M(i) → L(i) → 0

and

Corollary 6.14. Let 0 → N → M → L → 0 be an exact sequence of modules with a Δ (resp. ∇) filtration. Then for each 1 ≤ i ≤ m, the sequences h

k

h

k

0 → N (i) → M (i) → L(i) → 0,

are exact. 6.0.1. Tensor product of algebras. We leave to the reader to straightforward verification of Proposition 6.15. If A, B are two quasi hereditary algebras, so is A ⊗ B. The simple A ⊗ B−modules are of the form E ⊗ F with E a simple A-module and F a simple B-module. Similarly for the projective covers and injective hulls. From the definitions one then gets Δ(E ⊗ F ) = Δ(E) ⊗ Δ(F ),

∇(E ⊗ F ) = ∇(E) ⊗ ∇(F ).

Part III

The Schur algebra

50

III. THE SCHUR ALGEBRA

The study of the Schur algebra will be done using the combinatorial theory of semistandard bitableaux. This Theory is due to Doubilet–Rota–Stein [22] building on earlier work of Young, Hodge [29] and Igusa [31]. We shall follow the approach of Seshadri [58], [36]. 7. The Schur algebra We recall Definition 2.9 in the special case of the algebra A = End(V ) = V ⊗V ∗ of linear transformations on a free F -module V of rank m or, choosing a basis of V , of m × m matrices Mm (F ) with coefficients in the commutative ring F . Definition 7.1. The algebra SV,t := Sm,t := (A⊗t )St formed by the symmetric tensors in the algebra A⊗t is called a Schur algebra (see Green [27]). Recall that a polynomial representation U of degree t of GL(V ) extends to a polynomial map End(V ) → End(U ), homogeneous of degree t and which is multiplicative. By the universal property of Sm,t such a map factors thorough an algebra homomorphism Sm,t → End(U ), so: Theorem 7.2. There is a canonical equivalence, given by extension of the action, between the category of homogeneous polynomial representations of degree t of GL(m) and the category of Sm,t -modules. One approach to the study of the Schur algebra Sm,t is the following. Take the ring S[V ∗ ⊗ V ] = Rm = F [xi,j ], 1 ≤ i, j ≤ m of polynomial functions on A = End(V ) = V ⊗ V ∗ . On S[V ∗ ⊗V ] = Rm we have the polynomial action of GL(V ∗ )×GL(V ) induced by the natural action on V ∗ ⊗ V : (35)

(g, h)(φ ⊗ v) := gφ ⊗ hv,

(g, h) ∈ GL(V ∗ ) × GL(V ).

By duality the dual homF (Sm,t , F ) of Sm,t equals the space Rm,t of homogeneous polynomials of degree t in Rm . We define  xi,h ⊗ xh,j . (36) δ : Rm,1 → Rm,1 ⊗ Rm,1 , by δ(xi,j ) = h

δ induces a coassociative coproduct on Rm which we still denote by δ and, by homogeneity, δ(Rm,t ) ⊂ Rm,t ⊗ Rm,t . It is easy to see that its dual gives the associative product of the Schur algebra Sm,t . We finish with a remark which is going to be useful later on, op Lemma 7.3. SV ∗ ,t is canonically isomorphic to SV,t the opposite of the Schur algebra SV,t .

Proof. The anti-isomorphism is induced by the natural map i : V ⊗ V ∗ → V ∗ ⊗ (V ∗ )∗ = V ∗ ⊗ V, i(v ⊗ φ) = φ ⊗ v.



Our goal is to prove, when F is an infinite field, that the Schur algebra is quasi-hereditary, and to describe its ∇ modules, Theorem 9.36. This is done by a combinatorial method that of Double standard tableaux.

8. DOUBLE TABLEAUX

51

8. Double tableaux 8.1. Young diagrams, weights and tableaux. We start from a lattice ZN , with canonical basis ei . Notice that the symmetric group SN acts by permuting the coordinates.  N Definition 8.1. An element λ = (λ1 , λ2 , . . . , λN ) = is also i λi ei ∈ Z called a weight. A weight is called dominant if λ = (λ1 ≥ λ2 . . . ≥ λN ). Clearly any weight is conjugate to a unique dominant weight under the natural action of the symmetric group SN . N The integer t = i=1 λi is called the degree of λ and denoted |λ|.  In particular a dominant weight such that λN ≥ 0 with λj = t is a partition of t and we denote this by λ  t. Remark 8.2. A partition is also described by the function i → hi ∈ N where hi is the number of parts λs which are equal to i. Such a function determines a partition of the number i hi i. In this case we display the partition by the symbol: (37)

λ := 1h1 2h2 . . . khk .

We also display a partition as a diagram whose first column has λ1 boxes, the second λ2 and so on. Definition 8.3. The diagram corresponding to a partition λ is called the Young diagram of shape λ. We often use the notions of partition and Young diagram as synonymous. For example the Young diagram of shape λ = (3, 3, 2, 2, 1) = 11 22 32 is

Alternatively, the Young diagram of shape λ can be described by its dual partition ˇ In our previous example the dual partition is (5, 4, 2). of row lengths, denoted by λ. Given λ = (λ1 ≥ λ2 ≥ . . . ≥ λN ≥ 0) and μ = (μ1 ≥ · · · ≥ μN ≥ 0) two Young diagrams, we say that μ ⊂ λ if for every 1 ≤ j ≤ N , μj ≤ λj and we will sometime consider the skew diagram λ \ μ consisting of the boxes of λ which are not in μ. For example if λ = (3, 3, 3, 2, 1) and μ = (2, 1, 1, 0, 0) we get λ\μ=

There is a natural partial ordering on weights. It is a special case of the dominance order associated to root systems (see [30]). Definition 8.4 (Dominance order). Given two weights, λ = (λ1 , λ2 , . . . , λN ), and μ = (μ1 , μ we say that λ ≤ μ if they have the same degree and for 2 , · · · , μN ), every i ≤ N , j≤i λj ≤ j≤i μj .

52

III. THE SCHUR ALGEBRA

Exercise 8.5. (1) The elements αi,j := ei − ej , i < j are in fact the positive roots of the root system AN −1 , then one has λ ≤ μ if an only if  μ = λ + i 0) the canonical tableau Cλ of shape λ is the tableau of shape λ such that for each i = 1, . . . , N , each box in the i-th column is filled with the number i. Examples of canonical tableaux (here N = 4, 5): 1 2 3 4 1 2 1 2

1 2 3 4 5 1 2 3 1

We have Proposition 8.10. Let λ = λ1 ≥ . . . ≥ λN > 0 be a Young diagram and T a semistandard tableau of shape μ ≤ λ filled with the integers 1, 2, . . . , m, m ≥ N . Then 1) w(T ) ≤ w(Cλ ). 2) w(T ) = w(Cλ ) if and only if μ = λ and T = Cλ is the canonical tableau of shape λ. 3) There is a unique semistandard tableaux of shape λ filled with the integers 1, 2, . . . , m whose weight is lowest. Its weight is λr = {k1 , . . . , km } with ki = λm−i+1 . 4) In all the semistandard tableaux of shape λ filled with the integers 1, 2, . . . , N we have that N appears at least λN times. Proof. We have w(T ) = (k1 , . . . , km ) where ki is the number of occurrences of the index i. In a semistandard tableau the index i can only appear in the first i columns so, if μi is the length of the i-th column of μ, we must have k1 + . . . + ki ≤ μ1 +. . . +μi for all i which means w(T ) ≤ μ ≤ λ = w(Cλ ). The first two statements follow. As for the third observe that we can take the semistandard tableau in which each row with a boxes is filled with the last a indices m − a + 1, . . . , m. We leave to the reader to verify that this is the lowest possible weight. An index j ≤ m appears once in each row of length a with m − a + 1 ≤ j, that is a ≥ m − j + 1. The number of these rows is λm−j+1 . Finally, if m = N , since each i < N must be in one of the first i columns, it is  clear that the N th column must be occupied by the index N . 8.2. Tableaux as polynomials. Consider the space W  of -tuples v1 , . . . , v of elements of the free module W = F n , here F could be any commutative ring but for simplicity we assume F = Z or a field. We consider the vi as columns of an n ×  matrix so the polynomial ring of functions on the space W  is F [xi,j ], 1 ≤ i ≤ n, 1 ≤ j ≤ , where the xi,j are the

54

III. THE SCHUR ALGEBRA

entries of the generic matrix   x1,1  x X :=  2,1 ··· xn,1

x1,2 x2,2 ··· xn,2

x1,3 x2,3 ··· xn,3

··· ··· ··· ···

 x1,  x2,  . · · ·  xn, 

We assume that  ≥ n so that among these polynomials we can consider those which are multilinear and antisymmetric in exactly n of the column vectors vi . Of course a polynomial which is multilinear and antisymmetric in exactly n vectors vi1 , . . . , vin in F n is just a multiple of the determinant of the square matrix whose r-th column is vir , r = 1, . . . , n. In other words it is a multiple of the determinant of the n×n minor (of maximal order) of the matrix X extracted from the columns i1 , . . . , in . We shall denote this determinant as a row tableau pI :=

i1 i2 ..... in ,

where the index I = (1 ≤ i1 < · · · < in ≤ ). This allows us to introduce a partial ordering on these determinants by setting pI ≤ pJ

i.e.

i1 . . . in ≤ j1 . . . jn

if is ≤ js for all 1 ≤ s ≤ n. Notice that p0 := 1 ... n ≤ pI for all pI and the condition pI ≤ pJ is equivalent to state that:

(38)

i1 i2 . . . in

the tableau

is semistandard.

j1 j2 . . . jn Exercise 8.11. We have that two rows a := i1 . . . in , b := j1 . . . jn with a < b are adjacent (cf. Definition 8.6) if and only if in a there an index ir , which is either the last index in with in < , or if r < n, with ir < ir+1 , such that b is obtained from a by just replacing ir with ir + 1. As example the row tableau 1 2 4 has two adjacent tableaux (if  ≥ 5) namely 1

3

4 and 1

2

5 .

One should remark that this is truly the classical theory of Pl¨ ucker coordinates for the Grassmann variety of n-subspaces in an  dimensional space. The ordering is strictly connected with that of Sch¨ ubert cells. We will not pursue this (certainly fundamental) point of view since we want to keep the treatment combinatorial (but see [48]).

8. DOUBLE TABLEAUX

55

Example n = 2,  = 4, the poset in diagrammatic form: 3

4

2

4

ww ww w ww ww 1

HH HH HH HH H

4

2

GG GG GG GG G

v vv vv v v vv 1

3

1

2

3

We make now the simple, but basic, remark that a polynomial which is multilinear and in exactly n + 1 vectors in F n is necessarily equal to 0, that is n+1antisymmetric F n = 0. We apply this remark as follows (see also [68]). Consider the product of two determinants in disjoint vector variables. φ(v1 , . . . , v2n ) = |1, 2, . . . n||n + 1, n + 2, . . . 2n| Alternatively we can display φ as a Young tableau with two rows, Formula (38). For k with 1 ≤ k ≤ n consider the n + 1 vector variables vk , . . . , vn , vn+1 , . . . , vn+k . If we alternate the function φ in these variables, we get a polynomial function which, as a function of then n + 1 vectors vk , . . . , vn , vn+1 , . . . , vn+k , is multilinear and alternating and hence, by the previous remark, equal to 0. Since φ(v) is already alternating separately in vk , . . . , vn and in vn+1 , . . . , vn+k the alternation is obtained by summing with signs over left coset representatives with respect to the Young subgroup Sn−k+1 × Sk which permutes separately these two sets of variables. For instance, using the tableau notation for n = k = 2 we have 1 2 1 3 1 4 − + = 0. 3 4 2 4 2 3 Notice that the two tableaux 1 4 and 2 3 are the ones which are not comparable in our poset. Since these are symbolic identities they give other identities by specializing the vector variable vi to any other vector variables. In particular, take two sequences, I = (i1 , . . . , in ) and J = (j1 , . . . , jn ) an let pI and pJ denote the corresponding determinants. If two of the indices i (resp. j) are equal then pI = 0 (resp. pJ = 0). If they are distinct, up to possibly introducing a sign in pI and pJ , we can assume that the i’s and j’s are strictly increasing. Next display the product pI pJ as in Formula (38): pI pJ = i1 i2 . . . in . j1 j2 . . . jn

56

III. THE SCHUR ALGEBRA

Suppose is ≤ js for s < k while j1 < j2 < · · · < jk < ik < ik+1 < · · · < in . Then we can apply the above procedure and write pI pJ as a linear combinations (with coefficients ±1) of products pIr pJr with pIr < pI (and pJ < pJr ). Definition 8.12. A quadratic equation applied to pI pJ by alternating the n+1 indices i1 , . . . , ik , jk , . . . , jn is called a straightening relation. Let us illustrate our procedure by an example. Start with the tableau n = 4,

1

3

4

1

2

4

5 . 6

We then perform the alternation with respect to the 5 entries 3 1

4

5 .

2

In the resulting relation all terms in which we have brought 1 from the second to the first row vanish since they give a tableau with two equal indices, that is a determinant with two equal vectors, so only the 3 terms obtained by exchanging 2 with 3, 4, 5 remain. Of these the exchange of 2 and 4 brings two 4 on the second row so it also gives 0. In conclusion we get 1 3 4 5 1 2 4 5 = + 1 2 4 6 1 3 4 6

1 3 4 2 1 5 4 6

=

1 2 4 5 1 3 4 6



1 2 3 4 . 1 4 5 6

Definition 8.13. Let us denote by R the subring R ⊂ F [xi,j ] generated by the determinants of the maximal order, i.e. n × n, minors pJ of the matrix X. Remark 8.14. The determinants pJ are clearly invariant under the action of the group SL(n, F ) by left multiplication. In fact it can be proved (see Exercise 9.26) that R = F [xi,j ]SL(n,F ) is the full ring of invariants (a Theorem of Igusa [31]). Since any such minor is a homogeneous polynomial of degree n, R inherits a grading from the natural grading of F [xi,j ]. Since R ∩ (F [xi,j ])h = {0} only if h is a multiple of n, we write R = ⊕Rr , with Rr := R ∩ (F [xi,j ])nr . We may think any rectangular tableau of the form h1,1 h1,2 . . . . . . h1,n h2,1 h2,2 . . . . . . h2,n ... ... ... ... ... hr,1 hr,2 . . . . . . hr,n 1 ≤ hs,t ≤  as the element in pH1 pH2 · · · pHr ∈ Rr product of its rows. We may assume that the indices in each row are strictly increasing and we have Remark 8.15. A rectangular tableau is semistandard if and only if we have pH1 ≤ pH2 ≤ · · · ≤ pHr . If this is the case this will be called a standard monomial. Theorem 8.16. The standard monomials, in the elements pI ’s, with I = (1 ≤ i1 < · · · < in ≤ ), are a linear F basis of the ring R.

8. DOUBLE TABLEAUX

57

Proof. We first show that the standard monomials of degree r span Rr . For this it clearly suffices to show that any monomial pH1 pH2 · · · pHr is a linear combination of standard ones of the same degree. We proceed by induction on the partial ordering on the pI ’s and on r. If r = 1 there is nothing to show. So we may assume pH2 · · · pHr is standard. At this point if pH1 ≤ pH2 (in particular if pH1 = p0 ), also pH1 pH2 · · · pHr is standard and we are done. Otherwise, h1,s > h2,s for some 1 ≤ s ≤ n and we can write pH1 pH2 as a linear combination of two rowed monomials pH1 pH2 with pH1 < pH1 by applying the straightening relation of Definition 8.12. Substituting we have written pH1 pH2 · · · pHr as a linear combination of monomials pK1 pK2 · · · pKr which are either standard or have the property that pK1 < pH1 , so everything follows by induction. We now prove the linear independence of the standard monomials of a given degree. To pI = i1 ... in let us associate the ideal YI in F [xi,j ] generated by the variables xs,t with 1 ≤ s ≤ n, t < is and let II ⊂ R be its intersection with R. II is clearly a prime ideal, furthermore: Lemma 8.17. II ⊂ IJ if pI ≤ pJ and pJ ∈ II if and only if pI ≤ pJ . Proof. The first claim is clear since pI ≤ pJ means that is ≤ js for all s. As for the second claim, if pI ≤ pJ the variables xs,js are not in the ideal YI so clearly pJ = s xs,js + C is nonzero modulo YI . Now, if pI ≤ pJ we have for some s that js < is , then modulo YI , the first s columns of the minor associated to J have nonzero entries only in the first s − 1  rows, so they are linearly dependent and hence pJ = 0 modulo YI . We now show that the semistandard tableaux pH1 pH2 · · · pHr with pH1 ≥ pI are linearly independent modulo II thus finishing the proof of Theorem 8.16 First if I = ( − n + 1, . . . , ) we have pJ ≤ pI , ∀J so, by Lemma 8.17, R/II is the polynomial ring F [y], with y the class of pI modulo II and the claim follows. We now proceed by induction on the partial ordering. Suppose there is a linear combination u  at pH (t) pH (t) · · · pH (t) = 0 1

t=1

r

2

modulo II , with pH (t) pH (t) · · · pH (t) semistandard, at = 0, pH (t) ≥ pI for each t. r 1 2 1 If for some t, pH (t) > pI we can further quotient modulo IH (t) and use the 1 1 inductive assumption to get a contradiction. So we can assume pH (t) = pI for each 1 t and our relation becomes u  pI ( at pH (t) · · · pH (t) ) = 0. r

2

t=1

Now since R/II is a domain, we can divide by pI and deduce u  t=1

at pH (t) · · · pH (t) = 0. 2

r

At this point everything follows by induction on the degree r.



58

III. THE SCHUR ALGEBRA

Remark 8.18. The previous proof is a simplified version of a stronger Theorem, due to Seshadri [58], on the projective coordinate rings of Sch¨ ubert varieties. Let p = p(−n+1,...,) and notice that pH1 pH2 · · · pHr is standard if and only if the monomial pH1 pH2 · · · pHr p is standard. We have Corollary 8.19. 1) Let P denote the ideal of F [xi,j ] generated by p − α, with α ∈ F an invertible element of F (in particular α = ±1). Then P ∩ R = (p − α)R. 2) The standard monomials pH1 pH2 · · · pHr with pHr = p (together with 1) are a F -linear basis of R/(p − α)R. Proof. 1) It is clear that P ∩ R ⊃ (p − α)R. Now take (p − α)a = b ∈ R write  a = h ah and b = h bh with ah , bh homogeneous of degree h (under the grading / R and let h be of F [xi,j ]). Since R is graded, bh ∈ R for each h. Assume that a ∈ the minimum such that ah ∈ / R. We have bh = pah−n − αah where, if h − n < 0, ah−n = 0. It follows that ah = α−1 (pah−n − bh ) ∈ R. A contradiction. 2) Is clear since by Theorem 8.16, R is a free module over the polynomial ring F [p] with basis the semistandard rectangular tableaux pH1 pH2 · · · pHr with pHr = p (together with 1).  8.3. Double standard tableaux. Now, following Seshadri [36] we construct the theory of semistandard bitableaux, which originally is due to Doubilet, Rota, Stein [22]. The basic remark of Seshadri is the following. Take an n × m matrix X and construct from X the n × m + n matrix (X, Jn ), with    0 0 · · · 1    0 ··· 1 0  Jn :=  · · · · · · · · · · · ·   1 0 ··· 0  getting

   x1,1 x1,2 x1,3 · · · x1,m 0 0 · · · 1    x2,1 x2,2 x2,3 · · · x1,m 0 ··· 1 0   . ··· ··· ··· ··· ··· · · · · · · · · · · · ·  xn,1 xn,2 xn,3 · · · xn,m 1 0 ··· 0  If we now compute the determinant of a maximal minor, we have for instance      x1,2 x1,4 0 x1,2 0 1  x1,2 x1,4       , det x2,2 1 0 = −x3,2 det x2,2 x2,4 1 = − det     x3,2 x3,4 x3,2 x3,4 0 x3,2 0 0

we see that this equals, up to some sign, the determinant of some suitable minor of X, therefore we have a way to deduce from results on maximal minors also results on all minors. To be precise Lemma 8.20. The determinant of the maximal minor of (X, Jn ) extracted from the columns 1 ≤ j1 < j2 < . . . < jk ≤ m and the columns m < m + s1 < . . . < m + sn−k ≤ m + n equals, up to a sign, the determinant of the minor of X extracted from the columns 1 ≤ j1 < j2 < . . . < jk ≤ m and the rows 1 ≤ i1 < i2 < . . . < ik ≤ n with (39)

{i1 , . . . , ik } ∪ {n − sn−k + 1, . . . , n − s1 + 1} = {1, . . . , n}.

8. DOUBLE TABLEAUX

59

Proof. This is immediate from the Laplace expansion of the determinant, relative to the last n−k columns. The only nonzero determinant is the one extracted  from the rows {n − sn−k + 1, . . . , n − s1 + 1} hence the claim. Definition 8.21. A row bitableau of shape k is the determinant of a k × k minor of the matrix X. If the minor is extracted from rows 1 ≤ i1 < i2 < . . . < ik ≤ n and columns 1 ≤ j1 < j2 < . . . < jk ≤ m we denote it by ik . . . . . . . . . i2 i1 j1 j2 . . . . . . . . . jk We have thus established a 1–1 correspondence between all row tableaux of length n, filled with numbers ≤ m + n with the exception of the highest element which gives (−1)n+1 , and row bitableaux of shapes k ≤ n with the left tableau in the indices 1, . . . , n and the right in the indices 1, . . . , m. Thus this correspondence induces a partial order in the set of row bitableaux. We can analyze this using the Exercise 8.11 on adjacent row tableaux. Take two adjacent such bitableaux a, b corresponding to row tableaux a and b , with b

obtained from a replacing an index i by i + 1. Proposition 8.22. We have then 3 different cases. Assume that a is of shape k. (1) i + 1 ≤ m, then b is also of shape k. Moreover b is obtained from a by replacing, in its right tableau the index i with i + 1. (2) m + s = i > m, then b is also of shape k. Moreover b is obtained from a by replacing, in its left tableau the index h corresponding to i with h + 1. (3) i = m then b is of shape k − 1. Moreover b is obtained from a by removing in its left tableau the index m and in its right tableau the leftmost index which necessarily equals to n. Proof. Let us prove the last two statements. 2. If we are replacing some m + s with m + s + 1, using formula (39), this is possible only if among the indices i there is n − (s + 1) + 1 = ia and this is replaced by n − s + 1 = ia + 1. 3. Suppose a = j1 < j2 < . . . < jk = m < m + s1 < . . . < m + sn−k , which is associated to the bitableau with left tableau filled with j1 < j2 < . . . < jk = m and right tableau filled with the numbers i1 < i2 < . . . < ik defined by {s1 , . . . , sn−k } ∪ {n − ik + 1, . . . , n − i1 + 1} = {1, . . . , n}. Now since by hypothesis we can perform the move m → m + 1 we must have s1 > 1 which implies n − ik + 1 = 1 that is n = ik . Finally the new element b = j1 < j2 < . . . < jk−1 < m + 1 < m + s1 < . . . < m + sn−k is clearly obtained as claimed.  Definition 8.23. A bitableau of shape λ = k1 ≥ k2 ≥ . . . ≥ kr i1,k1 . . . . . . . . . i1,2 i1,1 j1,1 j1,2 . . . . . . . . . j1,k1 i2,k2 . . . . . . i2,2 i2,1 j2,1 j2,2 . . . . . . j2,k2 ... ... ... ...

... ... ... ...

ir,kr . . . ir,1 jr,1 . . . jr,kr

60

III. THE SCHUR ALGEBRA

is a product of r row bitableaux of shapes k1 ≥ k2 , . . . ≥ kr . It is called semistandard if both its right and it left tableaux are semistandard, that is is,t < is,t+1 and is,t ≤ is+1,t , js,t < js,t+1 and js,t ≤ js+1,t whenever this makes sense. Now take a monomial pH1 pH2 · · · pHr with pHj = p for each j = 1, . . . , r. To each pHj we associate a row bitableaux of shapes ki and up to reordering the factor, we can assume that k1 ≥ . . . ≥ kr so we get, up to a suitable sign, a bitableau product of the row bitableaux corresponding to each of the pHi (of course to p we associate (−1)n+1 ). By Proposition 8.22 this gives a bijection between the set of standard monomials pH1 pH2 · · · pHr , with pHr = p and the set of semistandard bitableaux. We may also treat the same objects in a second form. We illustrate this with an example. Let n = 5, m = 7 so m + n = 12 replace an index x > 7 with x − 7 but marked blue, the ones ≤ 7 marked in red: 1

3

5

6

9

2

3

6

7

2

3

7

8

4

5

9

1

3

5

6

2

9

2

3

6

7

2

10

2

3

7

1

3

11 12

4

5

2

4

5





5

4

2

1

1

3

5

6

5

4

2

1

2

3

6

7

5

3

1

2

3

7

4

2

4

5

Given m (in this case m = 7) these are 3 displays of the same object. The second display also presents the object as a pair of tableaux, but now not of the same but of complementary shapes. We shall see in the next section, cf. Proposition 9.16, the meaning of this as a suitable duality. 8.3.1. The straightening relation for bitableaux. We finally need to understand, in the language of bitableaux, the straightening relation of Definition 8.12. So let a, b be two row bitableaux of shapes h ≥ k. Assume that a corresponds to pI and b to pJ . We then write pI pJ as a linear combination of standard monomials pIs pJs with Is < I and J < Js going back to bitableaux we get that unless pIs = p, pIs pJs corresponds to a semistandard Young bitableau as bs with as , bs of shapes hs ≥ h ≥ k ≥ ks and h + k = hs + ks , while, if pJs = p, then pIs pJs corresponds to a single row bitableau of shape h + k. We deduce Proposition 8.24. Any bitableau T of shape λ is a linear combination of standard bitableaux whose shapes are less than or equal to λ. 8.3.2. The main theorem for bitableaux. We then can deduce form Theorem 8.16 the main theorem for bitableaux: Theorem 8.25. Let F be a commutative ring with 1. The polynomial ring F [xi,j ], 1 ≤ i ≤ n, 1 ≤ j ≤ m has as F linear basis the semistandard bitableaux whose left tableau has entries positive integers less than or equal n and whose right tableau has entries positive integers less than or equal m. Proof. Every monomial of degree k in the variables xi,j is a bitableau with shape a single column of length k. So, by Proposition 8.24 the semistandard bitableaux linearly span F [xi,j ]. In order to prove that they are linearly independent it is clearly enough to prove that in each degree they are as many as the dimension

8. DOUBLE TABLEAUX

61

of F [xi,j ] in that degree. Since this dimension is independent of F we may assume F = C where we can use a geometric argument. Let Y denote the affine space of complex n×(m+n) matrices, Z the affine space of n×m matrices and G = SL(n) the group of n×n matrices of determinant 1 which acts on Y by left multiplication. The algebra R is formed by G invariant elements (in fact it coincides with the G-invariants, by Exercise 9.26) of the coordinate ring O(Y) of the space Y. Consider the commutative diagram / Z ×G ZF FF FFS κ FF FF  " Y χ

Where χ(A) = (A, In ) ∈ Z × SL(n), with In the identity n × n matrix. While κ(A, g) = (gA, gJn ) and S(A) = (A, Jn ). The group G acts on Z × G by acting on the right factor by left multiplication. The coordinate ring O(Z × G) = O(Z) ⊗ O(G) and thus the coordinate ring O(Z) of Z is isomorphic, under the map χ∗ , to the ring of G invariant functions of this action, furthermore κ is G-equivariant. Notice that κ is a isomorphism onto the hypersurface in Y defined by the equation p − (−1)n+1 = 0. Since p − (−1)n+1 is an irreducible polynomial, the kernel of the map κ∗ is the ideal generated by p − (−1)n+1 . It thus follows from Corollary 8.19 1), that κ∗ (R)  R/(p − (−1)n+1 )R. Also R consists of G invariant polynomial, so since S = κ ◦ χ and χ∗ is injective on G invariants, we have: ker S ∗ ∩ R = ker(χ∗ ◦ κ∗ ) ∩ R = ker κ∗ ∩ R = (p − (−1)n+1 )R. Thus S ∗ induces an injection s : R/(p − (−1)n+1 ) → C[xi,j ], 1 ≤ i ≤ n, 1 ≤ j ≤ m. Here C[xi,j ] is the coordinate ring O(Z) of the space Z. Now notice that by Lemma 8.20, if I = (j, m + 1, . . . , m + n − i, m + n − i + 2, . . . m + n), s(pI ) = ±xi,j , so s is onto. Since we have already remarked that up to signs, the image under S of the set of standard monomials pH1 pH2 · · · pHr with pHr = p, maps bijectively onto the set of semistandard bitableaux whose left tableau has entries positive integers less than or equal n and whose right tableau has entries positive integers less than or equal m, everything follows from the second part of Corollary 8.19. 

62

III. THE SCHUR ALGEBRA

9. Modules for the Schur algebra In this section we interpret the theory of bitableaux from the point of view of representations of the Schur algebra, Sm,t  SV,t , dim V = m or of the linear group GL(m, F ) = GL(V ). 9.1. The action of the Schur algebra on polynomials. We consider a generic matrix ξ with entries the variables xi,j , where, in general, we assume i = 1, . . . , n, j = 1, . . . , m. For a given coefficient ring F , which we take to be either the integers or an infinite field, we have that the polynomial ring F [xi,j ], i = 1, . . . , n, j = 1, . . . , m, is the ring of polynomial functions on n × m matrices over F . In more intrinsic terms, such matrices are hom(V, W ) where V is an m-dimensional (with basis e1 , . . . , em ) and W an n-dimensional (with basis f1 , . . . , fn ) space. So F [xi,j ] is the ring of polynomial functions on hom(V, W ) = W ⊗ V ∗ or the symmetric algebra over its dual. So we have (40)

F [xi,j ] = S[hom(V, W )∗ ] = S[W ∗ ⊗ V ],

xi,j = f i ⊗ ej .

As a function of A ∈ hom(V, W ) we have (41)

xi,j : A → f i | Aej .

The two linear groups GL(W ) × GL(V ) = GL(n, F ) × GL(m, F ) act on the space of matrices by (g, h)C := gCh−1 , (g, h) ∈ GL(W ) × GL(V ). This action induces an action on the polynomial ring by (42)

(g, h)f (C) = f ((g, h)−1 C) = f (g −1 Ch).

We see that, on the span on the variables xi,j the group GL(V ) acts as a homogeneous representation of degree 1, that is n copies of V , while GL(W ) acts as a homogeneous representation of degree -1, that is m copies of W ∗ . Indeed, cf.  n−1 W ⊗ det−1 (where by det we mean the 1 dimensional represen§2.1, W ∗ = dim W tation W and det−1 its dual) it is the tensor product of a homogeneous representation of degree n − 1 times det−1 , so it is homogeneous of degree −1. It is useful to think of S[W ∗ ⊗ V ] as a polynomial representation of the group GL(W ∗ ) × GL(V ). On the span on the variables xi,j the group GL(W ∗ ) × GL(V ) acts as a homogeneous representation of degree 1. Of course we have a canonical isomorphism GL(W ) ∼ = GL(W ∗ ) by mapping g → (g −1 )∗ . Of particular importance is the case W = V . In this case as we have already seen in Section 7, S t (hom(V, V )∗ ) = Rm,t is the dual of the Schur algebra SV,t . The two actions of GL(V ) × GL(V ) given by Formula (42) give rise, through the isomorphism g → (g −1 )∗ to two polynomial actions of GL(V ∗ ) and GL(V ), hence to two actions of the Schur algebras SV ∗ ,t and SV,t and we have: Proposition 9.1. (1) The action of SV,t coincides, under the identifica∗ with the dual of the right action. tion S t (hom(V, V )∗ ) = SV,t ∗ and (2) The action of SV ∗ ,t , under the identification S t (hom(V, V )∗ ) = SV,t op SV ∗ ,t = SV,t (cf. Lemma 7.3) coincides with the dual of the left action. Proof. 1). It is enough to show that the two actions coincide for decomposable elements g ⊗t ∈ SV,t , g ∈ GL(V ) ⊂ End(V ). In this case the right action of g ⊗t maps a decomposable element C ⊗t to ⊗t ⊗t C g = (Cg)⊗t and induces on the dual φ | (Cg)⊗t  = f (Cg),

f (C) = φ | C ⊗t .

9. MODULES FOR THE SCHUR ALGEBRA

63

The proof of 2) is similar. Under the two identifications, for h ∈ GL(V ), we have j : h → (h−1 )∗ → h−1 and φ | j(h)C ⊗t  = f (h−1 C).



9.2. Young bitableaux and the GL(W ) × GL(V ) action. We want to describe how the basis of S[W ∗ ⊗ V ] consisting of Young bitableaux behaves under the G = GL(W ) × GL(V ) action. Of course we assume that we have chosen bases for W and V so that G = Gl(n, F )×Gl(m, F ) and S[W ∗ ⊗V ] = F [xi,j ], i = 1, . . . , n, j = 1, . . . , m. We start with the action of diagonal matrices. 9.2.1. Combinatorial theory of weights. When F is an infinite field the content of a tableau (or of a bitableau) has an interpretation in terms of eigenvalues of diagonal matrices. As in Proposition 2.16, can identify the lattice of weights ZN with the character group of the torus DN = (F ∗ )N . Take the subgroups Dn ⊂ GL(W ) and Dm ⊂ GL(V ) of diagonal matrices. Given (t¯, s¯) = ((t1 , . . . , tn ), (s1 , . . . , sm )) ∈ Dn × Dm , a bitableau T whose left tableau has content (h1 , . . . , hn ) and whose right tableau has content (k1 , . . . , km ) transforms as (t¯, s¯) T =

n 

i t−h i

i=1

n 

ski i T.

i=1

Definition 9.2. The weight of a bitableau T = (T | Tr ) is the pair w(T ) = (−w(T ), w(Tr )). Let us point out the immediate but important Lemma 9.3. A bitableau T of shape λ is a linear combination of semistandard bitableaux of weight w(T ) and of shape μ ≤ λ. Proof. This is clear from the nature of the straightening relations and Proposition 8.24.  9.2.2. The action on row bitableaux. Let us assume k ≤ min(n, m). We have a homogeneous polynomial map wk : hom(V, W ) → hom(

k 

V,

k 

W ), wk : A →

k 

A.

Since wk has degree k, this induces a linear map wk∗ : (hom(

k 

V,

k 

W ))∗ → S k (hom(V, W )∗ ) = S k (W ∗ ⊗ V ).

Lemma 9.4. The map wk∗ is a G = GL(W ) × GL(V ) equivariant isomorphism     of hom( k V, k W )∗ = k W ∗ ⊗ k V with the span of row bitableaux of length k. Proof. For the chosen bases e1 , . . . , em of V and f1 , . . . , fn of W with dual basis f 1 , . . . , f n , we get, by formula (41) for the generic matrix ξ = (xi,j ), ξej =

n  i=1

xi,j fi with xi,j = f i | ξej  ∈ hom(V, W )∗ = W ∗ ⊗ V.

64

III. THE SCHUR ALGEBRA

By Remark 2.1, the element f i1 ∧ f i2 ∧ . . . ∧ f ik ⊗ ej1 ∧ ej2 ∧ . . . ∧ ejk gives the polynomial (in ξ): f i1 ∧f i2 ∧. . .∧f ik | (43)

k 

ξ(ej1 ∧ej2 ∧. . .∧ejk )=f i1 ∧f i2 ∧. . .∧f ik |ξej1 ∧ξej2 ∧. . .∧ξejk  

= det(xis ,jt ) = ik . . . . . . . . . i2 i1 j1 j2 . . . . . . . . . jk We then have the immediate consequence.

Proposition 9.5. The linear span Tλ of bitableaux of some given shape λ is stable under the action of the group G = GL(W ) × GL(V ). Proof. Since G acts as algebra automorphisms, this follows from Lemma 9.4.  Corollary 9.6. The linear span Tλ of bitableaux of shape λ and row shape ˇ = k1 , . . . , kr as representation of the group GL(W ) × GL(V ) is a homomorphic λ image of (

k1 



W ⊗

k1 

V)⊗(

k2 



W ⊗

k2 

V ) ⊗ ... ⊗ (

kr 



W ⊗

kr 

V ).

If we set |λ| := k1 + . . . + kr we have that Tλ is a polynomial representation of GL(V ) homogeneous of degree |λ|. On the other hand as GL(W ) representation it is of degree −|λ|.  Lemma 9.4 also allows us to get an explicit copy of k V inside S k (W ∗ ⊗ V ).  To do this, let first define the embedding ik : k V → V ⊗k by  (44) ik (v1 ∧ v2 ∧ . . . ∧ vk ) = σ vσ(1) ⊗ . . . ⊗ vσ(k) . σ∈Sk

We then embed V ⊗k into S k (W ∗ ⊗ V ) by v1 ⊗ v2 ⊗ . . . ⊗ vk →

k 

f i ⊗ vi .

i=1

Composing the two embeddings which are both GL(V ) equivariant, we clearly get, Proposition 9.7. The span of all left canonical bitableaux of the form k . . . . . . . . . 2 1 j1 j2 . . . . . . . . . jk k is a GL(V ) submodule isomorphic to V. Remark 9.8. Observe that the map ik is NOT a lift of the natural quotient map v1 ⊗ v2 ⊗ . . . ⊗ vk → v1 ∧ v2 ∧ . . . ∧ vk , defining ∧k V . In fact this has a splitting only if the characteristic does not divide k! in which case the splitting is 1  v1 ∧ v2 ∧ . . . ∧ vk → σ vσ(1) ⊗ . . . ⊗ vσ(k) . k! σ∈Sk

Moreover provided the characteristic is = 2 we can characterize: (45)

k 

V = {u ∈ V ⊗k | σu = σ u, ∀σ ∈ Sk }.

9. MODULES FOR THE SCHUR ALGEBRA

65

9.3. The Schur modules ∇λ (V ). Observe that, if M is any GL(W )×GL(V ) module and χ is a character of the diagonal subgroup D of GL(W ) the subspace M χ of elements of M transforming under D with χ is a GL(V ) module. We now take n = dim W ≥ m = dim V . In particular this allows us to give the following main definition: Definition 9.9. [Schur module] For λ = (λ1 ≥ . . . ≥ λm ≥ 0) a given Young diagram, the Schur module ∇λ (V ),3 is the GL(V ) module spanned by all bitableaux of shape λ whose left tableau is canonical. h Proposition 9.10. (1) ∇1h (V ) is isomorphic to V. (2) ∇λ (V ) has as basis the semistandard bitableaux of shape λ whose left tableau is canonical. (3) There is a surjective GL(V ) equivariant map (46)

πλ :

k1 

V ⊗

k2 

k h1

V ⊗ ...⊗



V → ∇λ (V )

ˇ are the row lengths of the diagram λ. where (k1 , k2 , . . . , kh1 ) = λ Proof. (1) is just Proposition 9.7. (2) By Lemma 9.3 a bitableau of shape λ whose left tableau is canonical is a linear combination of semistandard bitableaux of the same weights whose shape μ is less than or equal to λ. But by Proposition 8.10 any such a bitableau is necessarily of shape λ with canonical left tableau giving (2). h (3) follows from part (1), as we identify ∇1h (V ) with V , and the definition of a bitableau as a product of its rows.  Remark 9.11. We have already noticed that one may consider the algebra S[W ∗ ⊗ V ] = S[(V ∗ )∗ ⊗ W ∗ ] as polynomials on hom(W ∗ , V ∗ ) = hom(V, W ),

x → x∗ .

The formula on polynomials is f → f ∗ ,

f ∗ (x) = f (x∗ ).

This has as basis the same semistandard bitableaux but exchanging the right with the left tableau. Thus the span of all tableaux T of shape λ = (s1 , . . . , sn ) such that the right tableau is canonical has as basis the semistandard bitableaux T of shape λ with the right tableau canonical and it equals ∇λ (W ∗ ). Remark 9.12. Since ∇λ (V ) has a basis of semistandard bitableau for which the left tableau is fixed and canonical, we often ignore it and speak of the basis of ∇λ (V ) by semistandard tableaux (not double). The fact that the left tableau is canonical is not essential. We would have the same module (up to canonical isomorphism) also if the left tableau is obtained from the canonical one applying a permutation σ ∈ Sn to its entries. If σ is the permutation (n, n − 1, . . . , 1) the corresponding left tableau will be called anticanonical. Remark 9.13. One should remark that, by Definition 9.9, the module ∇λ (V ) is really defined over Z that is when V = Zm we have ∇λ (Zm ) ⊂ Z[xi,j ]. Then for any field F one has, for V = F m that ∇λ (V ) = ∇λ (Zm ) ⊗ F . 3 We shall prove in Theorem 9.36 that in fact this is a ∇ module for the Schur algebra S V,|λ| which is quasi-hereditary.

66

III. THE SCHUR ALGEBRA

The action of GL(m) should be understood as in the formal approach of §2.4 by the natural coaction (on the variables or the tableaux). We need to distinguish the variables appearing in the tableaux with the coordinates of GL(m) which we thus set to be yi,j (compare with Formula (36)): δ : ∇λ (Zm ) → ∇λ (Zm ) ⊗ Z[yi,j ][d−1 ] ⊂ Z[xi,j , yi,j ][d−1 ], induced by the polynomial action (as automorphisms) on the algebra Z[xi,j ] by δ : Z[xi,j ] → Z[xi,j , yi,j ] ⊂ Z[xi,j , yi,j ][d−1 ];

δ(X) = XY.

9.3.1. Cartan multiplication. Notice that the product of two left canonical bitableaux of shapes λ = (λ1 , . . . , λm ) and μ = (μ1 , . . . , μm ) is a left canonical bitableau of shape λ + μ = (λ1 + μ1 , . . . , λm + μm ). We thus have a surjective bilinear map ∇λ (V ) ⊗ ∇μ (V ) → ∇λ+μ (V ),

(47)

called Cartan multiplication, a reformulation of the map of Formula (46). In particular we have that the subspace ⊕λ ∇λ (V ) ⊂ S[W ∗ ⊗V ] is a subalgebra. We shall see in fact in Theorem 9.25 that this is the algebra of elements invariant under the unipotent group of strictly upper triangular matrices in GL(W ) 4 . In a completely analogous fashion we can consider the subalgebra having as basis the semistandard bitableaux whose right tableau is canonical. At this point we specialize and let W = V , so that S[V ∗ ⊗ V ] is the polynomial part of the coordinate ring of GL(V ). In particular n = m and the canonical bitableau of shape 1m corresponds to the determinant. In this case clearly the Cartan multiplication ∇1m (V )⊗∇μ (V ) → ∇1m +μ (V ) is just multiplication by the determinant and so it is an isomorphism. This suggests Definition 9.14. Given a dominant weight λ ∈ Zm , let k be the minimum non negative integer such that k1m + λ is a Young diagram. We set ∇λ (V ) := ∇k1m +λ (V ) det −k . 9.3.2. Schur functors. The construction of ∇λ (V ), for λ a Young diagram, is functorial in V , in fact, if f : U → V is a linear map we claim that this induces a map ∇λ (f ) : ∇λ (U ) → ∇λ (V ) compatible with the maps wλ . Indeed, take an auxiliary space W (of dimension ≥ ht(λ)) we have an induced map 1⊗f : W ∗ ⊗U → W ∗ ⊗V inducing a algebra homomorphism 1⊗f : S[W ∗ ⊗U ] → S[W ∗ ⊗V ]. It is then enough to observe that on row bitableaux of shape k this induces the map 1 ⊗ ∧k f so that k k U = ∇1k (U ) to V = ∇1k (V ). it maps 9.3.3. Duality. It is important to consider also ∇λ (V ∗ ) as a representation of GL(V ) under the isomorphism GL(V ) ∼ = GL(V ∗ ), X → (X ∗ )−1 . Since under this isomorphism the determinant maps to the inverse, it is enough to analyze the case when ∇λ (V ∗ ) is a homogeneous polynomial representation of GL(V ∗ ) of degree t := |λ| (Proposition 2.19 (2)). ˜ Let m := dim V and λ = λ1 ≥ λ2 ≥ . . . ≥ λm ≥ 0. Consider the partition λ ˜ with λi := λ1 − λm−i+1 . In diagram notation the Young diagram of λ is contained ˜ is the complement of λ in this rectangle, but it in a rectangle of sizes λ1 × m and λ ˜ appears flipped vertically and |λ| = λ1 m − |λ|. 4 This

is a combinatorial approach to the Borel–Weil theorem, [8] or [34].

9. MODULES FOR THE SCHUR ALGEBRA

◦ ◦ ◦ ◦ ◦ ◦ ˜ = 4, 3, 2, 0: Example m = 4, λ = 4, 2, 1, 0, λ ◦ ◦ ◦ ◦ ◦ ◦ ˇ of λ, then Notice that, if k1 , . . . , kr is the row partition λ ˜ is m − kr , . . . , m − k1 . λ

67

◦ ◦ ◦ ◦ the row partition of

In order to understand ∇λ (V ∗ ) let us consider it as submodule of S[V ∗ ⊗ V ] (that is we take W = V ). In End(V ) = V ⊗ V ∗ we have the affine open set GL(V ) defined by det = 0, so the coordinate ring A[GL(V )] on GL(V ) is the localized algebra A[GL(V )] := S[V ∗ ⊗ V ][det −1 ]. The generic matrix ξ is to be viewed as a matrix with entries in A[GL(V )]. This algebra is a Hopf algebra due to the group structure. In particular consider the antipode map S : A[GL(V )] → A[GL(V )] defined by Sf (g) := f (g −1 ) and characterized by S(ξ) = ξ −1 . m−1 ξ · det −1 . Recall that we have ξ −1 = Lemma 9.15. ak

...

a1

j1

...

jk

= ±S( im−k . . .

i1

b1

. . . bm−k ) det(ξ)

where the indices im−k , . . . , i2 , i1 resp. b1 , b2 , . . . , bm−k are complementary to the indices j1 , j2 , . . . , jk resp. ak , . . . , a2 , a1 in 1, 2, . . . , m. The sign is the product of the two signs of the permutations j1 , . . . , jk , i1 , . . . , im−k and a1 , . . . , ak , b1 , . . . , bm−k . Proof. From Formula (6) p(ξej1 ∧ . . . ∧ ξejk ) = j1 ,...,jk ξei1 ∧ . . . ∧ ξeim−k ⊗ det(ξ)u. We deduce p(ξej1 ∧ . . . ∧ ξejk ) = = = j1 ,...,jk = j1 ,...,jk

 



= j1 ,...,jk



ak . . . a1 j1 . . . jk p(ea1 ∧ . . . ∧ eak )

ak . . . a1 j1 . . . jk a1 ,...,ak eb1 ∧ . . . ∧ ebm−k ⊗ det(ξ)u eb1 ∧ . . . ∧ ebm−k | ξei1 ∧ . . . ∧ ξeim−k eb1 ∧ . . . ∧ ebm−k ⊗ det(ξ)u

ξ −1 eb1 ∧ . . . ∧ ξ −1 ebm−k | ei1 ∧ . . . ∧ eim−k eb1 ∧ . . . ∧ ebm−k ⊗ det(ξ)u.



S( im−k . . .

i1

b1

. . . bm−k )eb1 ∧ . . . ∧ ebm−k ⊗ det(ξ)u.

Hence ak

...

a1

j1

...

jk

= ±S( im−k . . .

i1

b1

. . . bm−k ) det(ξ). 

Proposition 9.16. If λ = (λ1 , . . . , λm ) is a partition, the module ∇λ (V ∗ ), dim V = m, as a representation of GL(V ) is isomorphic to (cf. Proposition 8.10, 3)) ∇λ˜ (V ) ⊗ d−λ1 = ∇−λr (V ), −λr = (−λm , . . . , −λ1 ).

68

III. THE SCHUR ALGEBRA

Proof. We use the antipode S which maps the elements xi,j entries of the matrix ξ to the elements yi,j entries of the matrix ξ −1 . By Lemma 9.15 we have that S maps the space spanned by k × k determinants into the space spanned by (m − k) × (m − k) determinants multiplied by d−1 . Moreover it maps the bitableaux whose left tableau is canonical of shape λ into the bitableau whose ˜ times det−λ1 and thus establishes a linear right tableau is anticanonical of shape λ ∗ ˜ i := λ1 − λm−i we have isomorphism between ∇λ (V ) and ∇λ˜ (V ) ⊗ d−λ1 . Since λ m r ˜ −λ1 1 + λi = −λ . The antipode exchanges the left and right actions, thus S(f g )(ξ) = f g (ξ −1 ) = f (ξ −1 g) = f ((g −1 ξ)−1 ) = Sf (g −1 ξ) = g (Sf )(ξ) So, the given isomorphism is a isomorphism of GL(V )-modules.



Remark 9.17. Remark that the given isomorphism carries semistandard bitableaux into analogous bitableaux, which are semistandard according to the reverse ordering, multiplied by d−r . 9.3.4. Standard filtration. The degree t homogeneous component S t [W ∗ ⊗ V ] is naturally a module for the Schur algebra SW ∗ ,t and SV,t and since the two algebras commute, a module for the algebra SW ∗ ,t ⊗ SV,t . Definition 9.18 (Standard filtration). Given a Young diagram λ  t we set Mλ ⊂ S t [W ∗ ⊗ V ] to be the subspace spanned by the bitableaux of shape less than or equal to λ. From Proposition 8.24 it follows that Theorem 9.19. (1) Mλ has as basis the semistandard bitableaux of shape less or equal than λ. (2) Mλ is a polynomial representation of GL(W ∗ )×GL(V ) of degree t. Equivalently Mλ is a SW ∗ ,t ⊗ SV,t -submodule of S t [W ∗ ⊗ V ].  Consider now the subspace Mλ = μ 0. For each i = 1, . . . , s, set Ti equal to the bitableau obtained from Ti substituting each b in the a-th column of the right tableau of Ti with a. Remark that Ti is semistandard and that furthermore if i = j, Ti = Tj . s It follows that (1+xea,b )Y has degree h, with leading coefficient i=1 ±ci Ti = 0 so (1 + xea,b )Y = Y. 2) This is an immediate consequence of 1) and Lie–Kolchin’s theorem (cf. [6] Chapter III), since U + acts as strictly upper triangular matrices on the entire polynomial ring. In fact the reader can verify that it also follows directly from the proof of 1) by a simple induction. Observing that, if Y ∈ A \ B, then the leading coefficient Y of (1 + xea,b )Y is in A and wr (Y ) < wr (Y ), after finitely many steps one has the desired element Y ∈ A ∩ B. 3) A Schur module ∇λ (V ) contains a unique canonical tableau so, by 2), every non zero submodule M ⊂ ∇λ (V ) contains the module generated by this canonical tableau. The first claim follows. The remaining statement follows by finding the least integer k such that k1m + λ is a Young diagram and writing ∇λ (V ) = ∇k1m +λ (V ) ⊗ det−k 4) and 5) have been shown in Proposition 8.10. 6) An irreducible module N can be embedded in the coordinate ring Rm [d−1 ] of GL(m) = GL(V ) by Theorem 2.17. By multiplication by some power of the

9. MODULES FOR THE SCHUR ALGEBRA

71

determinant d, N dk injects in Rm and its image is also irreducible. Thus it suffices to see that an irreducible module N in Rm is isomorphic to Iλ (V ) for some λ. Since N is irreducible there is a t such that N ⊂ Rm,t and hence we may assume that there exists a λ  t such that N ⊂ Mλ and N ⊂ Mλ . Then N , being irreducible, embeds in Mλ /Mλ , which by Corollary 9.20 is isomorphic to a sum of copies of ∇λ (V ) with socle a sum of copies of Iλ (V ). This immediately implies that M  Iλ (V ). Finally, if λ = μ, Iλ (V ) is not isomorphic to Iμ (V ), since the two modules have different highest weight.  Exercise 9.26. Prove that the subring R ⊂ F [xi,j ] generated by the determinants of the maximal order n × n minors pJ of the matrix ξ is the full ring of invariants: R = F [xi,j ]SL(n,F ) . Hint Consider the invariants under U − , and then exploit the fact that an invariant under SL(n, F ) is invariant under U + and U − . Remark 9.27. In characteristic 0, using the fact that GL(m) is linearly reductive one has that Iλ (V ) = ∇λ (V ) and one may recover the classical theory. Take W = V and consider the inclusion S[V ∗ ⊗ V ] ⊂ S[V ∗ ⊗ V ][d−1 ] = ¯ + denotes the subgroup of A[GL(V )]. One sees easily from Definition 9.9 that, if U ∗ strictly upper triangular matrices in GL(V ) which, under the canonical isomorphism with GL(V ) corresponds to U − : ¯+

Corollary 9.28. i) A[GL(V )]U = ⊕λ ∗ r ii) I−λ (V ) = Iλ (V ) .

dominant ∇λ (V

).

Proof. The first statement is dual to Theorem 9.25 1), let us prove the second. By Proposition 8.10 3), we have that the lowest weight of ∇λ (V ) is in fact λr and the unique weight vector of this weight is obtained from the highest weight vector by applying the element s0 = (m, m − 1, . . . , 1) ∈ Sm ∈ GL(V ). Thus this is also a vector in Iλ (V ). It follows, by Remark 2.20 and Definition 8.4, that Iλ (V )∗ has highest weight r  −λ so by part 3) it is isomorphic to I−λr (V ). From Proposition 8.10 it also follows: Corollary 9.29. The irreducible factors of a composition series of ∇λ (V ) different from Iλ (V ) are all of the form Iμ (V ) for some μ < λ. Repeating our analysis for the group of strictly upper triangular matrices in GL(W ∗ ) one can easily deduce the following corollary whose proof we leave to the reader. Corollary 9.30. Let m = dim V, the module ∇λ (V ) is isomorphic to the submodule of the coordinate ring F [GL(V )] = S[End(V )∗ ][det−1 ] = S[V ∗ ⊗ V ][det−1 ] which under the left action of B + (m) = T  U + (m) ⊂ GL(V ) transforms under the 1 −h2 m t2 . . . t−h character χ−h of B + (m). Here χ−h is the character of T given by t−h n 1 where h := h1 ≥ h2 ≥ . . . ≥ hm ≥ 0 are the columns of the partition λ. Notice that the proof of 1) of Theorem 9.25 in fact shows a stronger result which will be needed later when we analyze the branching rules. Let U + (i) be the subgroup of the unipotent group U + generated by all the root subgroups Ua,b , a < b ≤ i.

72

III. THE SCHUR ALGEBRA

Let us define a tableau T to be i-canonical if all indices j ≤ i which appear in T appear in its j th column, then: Theorem 9.31. For i ≤ m consider the subalgebra Bi ⊂ S[W ∗ ⊗ V ] spanned by the bitableaux whose right tableau is i-canonical. Then Bi is the subalgebra of U + (i) ⊂ GL(V )-invariant polynomials in S[W ∗ ⊗ V ]. 9.4. Schur modules and ∇-filtrations. 9.4.1. Some explicit injective modules. From now we are going to take W = V ∗ . and as usual use the given basis of V , to identify the algebra S[W ⊗ V ] with the algebra Rm = F [xi,j ], i, j = 1, . . . , m and GL(V ) with GL(m, F ). Furthermore for any Young diagram λ we are going to set ∇λ := ∇λ (V ), Δλ := Δλ (V ) = ∇λ (V ∗ )∗ and Iλ := Iλ (V ) unless otherwise specified. We have seen that the space Rm,t of homogeneous polynomials of degree t is dual to the Schur algebra Sm,t , hence every direct summand on this as left (or right) module is injective as a Sm,t -module. We look at some special left submodules. Given a vector μ = (μ1 , . . . , μm ) ∈ Nm with t = μ1 +· · ·+μm , define Rm,t (μ) to be the subspace of Rm,t spanned by all double tableaux with left tableau of content μ and hence of weight −μ (under left action). It is then clear that each Rm,t (μ) is GL(m, F ) module under the right action and it follows immediately that: Lemma 9.32. Rm,t is the direct sum of the spaces Rm,t (μ) which are thus injective left modules under the Schur algebra induced by the right action. Of course if λ1 , λ2 are conjugate under the symmetric group the two modules Rm,t (λ1 ) and Rm,t (λ2 ) are isomorphic. Since every weight is conjugate to a dominant weight we may consider in particular a dominant weight λ, that is a Young diagram. ˜ λ := Rm,t (λ). The module Rm,t (λ) has as basis We denote this injective by Q all the double standard tableaux whose left tableau has content λ. Among these there are the ones of shape λ, for these, by 8.10, the left tableau is canonical and they form a basis of the Schur module ∇λ defined in 9.10. ˜ λ is not the injective hull of ∇λ and the full description of the In general Q injective hull is quite dependent on the characteristic of the field and not very explicitly understood. Of course in characteristic 0 or very large the Schur algebra is semisimple and ∇λ is irreducible. In this case all modules are injective and projective. Observe now that by the straightening algorithm, and Proposition 8.10 ˜ λ has as basis all the double standard tableaux such Lemma 9.33. The module Q that its left tableau has content λ. Since λ is dominant it follows that the shape γ of each of these tableaux is γ ≥ λ. We have Theorem 9.34. Let λ = (h1 , . . . , hm ) be a Young diagram with t boxes. (1) Denote by Qλ the injective hull of ∇λ .   ∇λ = ker φ = ker φ. φ:Qλ →Qμ , μλ

(2) EndSm,t (∇λ ) = F .

φ:Qλ →Qμ , μ>λ

9. MODULES FOR THE SCHUR ALGEBRA

73

˜ λ factors through the Proof. (1) We clearly have that the embedding ∇λ ⊂ Q ˜ λ = Qλ ⊕ Q for some module Q . injective hull, and then, by injectivity we have Q λ λ So it is clearly enough to show that ∇λ is the intersection of all the kernels of all ˜λ → Q ˜ μ , μ  λ. homomorphisms φ : Q ˜ ˜ Let φ : Qλ → Qμ , μ a dominant weight, be such a homomorphism. If φ is non zero on ∇λ the image of ∇λ is a submodule and hence, by Theorem 9.25, it contains a canonical tableau, which is necessarily of content γ ≥ μ by Lemma 9.33. This canonical tableau must be the image of an element of weight γ in ∇λ . These elements are in the span of the standard tableaux of weight γ. Now we have seen that a weight which appears in ∇λ is ≤ λ so μ ≤ γ ≤ λ. Thus  ker φ. ∇λ ⊆ φ:Qλ →Qμ , μλ

˜ μ, μ > λ ˜λ → Q Now we need to show that the intersection of all the kernels φ : Q equals ∇λ . For this we us the action of GL(V ∗ ) × GL(V ) on Rm,t . Take the + subgroup U ⊂ GL(V ∗ ) of strictly upper triangular matrices and remark that + U ˜ λ ∩ Rm,t ∇λ = Q . +

The coordinate ring F [U ] equals the polynomial ring F [ti,j ], 1 ≤ i < j ≤ m + where ti,j is the function giving the i, j entry of a matrix in U . ¯ + on Rm,t induces a GL(V ) equivariant map So the action of U ˜ λ → Rm,t [ti,j ]. Γ:Q Composing with the projection to the polynomials Rm,t [ti,j ]+ without constant term, we get a map ˜ λ → Rm,t [ti,j ]+ Γ : Q whose kernel is the set of invariants and so by the previous remark ∇λ . In order to finish, is enough to observe that the coefficient of each positive it a degree monomial i,j ti,ji,j , in the image of Γ , lies in one of the injective modules  Rm,t (ν) where ν = λ + i λ so that Γ gives rise to an exact sequence 

Γ ˜ kμμ , ˜ λ −−− 0 −−−−→ ∇λ −−−−→ Q −→ ⊕ν>λ Rm,t (ν)[ti,j ] ∼ = ⊕μ>λ Q

as desired. (2) Let L ⊂ ∇λ denote the 1-dimensional subspace spanned by the canonical tableau. L is the subspace of vectors of weight λ. Thus for any φ ∈ EndSm,t (∇λ ), the restriction of φ to L is the multiplication by a scalar ζ ∈ F . We claim that φ = ζ Id. Indeed L ⊂ Ker(φ − ζ Id) and Im(φ − ζ Id) does not contain L. However if φ − ζ Id = 0, its image is a non zero submodule which by Theorem 9.25.2 contains L. Necessarily φ = ζId and everything follows.  Remark 9.35. A completely analogous statement to Theorem 9.34 holds for opp using the modules ∇λ (V ∗ ). We leave the details to the reader. the algebra Sm,t By Theorem 9.25 the correspondence λ → Iλ allows to identify the set Yt,m of Young diagrams with t boxes and at most m columns with the set of simple Sm,t -modules. Let us fix a total order on Yt,m which refines the dominant order. We get

74

III. THE SCHUR ALGEBRA

Theorem 9.36. The algebra Sm,t with the chosen order is quasi-hereditary. Furthermore for each λ ∈ Yt,m , ∇(λ) = ∇λ and Δ(λ) = Δλ . Proof. The fact that ∇(λ) = ∇λ is just the first assertion of Theorem 9.34. Similarly, by Remark 9.35 the modules ∇λ (V ∗ ) are the ∇ modules for the opp with respect to the chosen order so, by duality, Δ(λ) = Δλ . algebra Sm,t To show that Sm,t with the chosen order is quasi-hereditary, by Definition 6.10, we need to see that Sm,t has a Δ-filtration and that for each λ ∈ Yt,m , EndA (Δ(λ)) = F. opp -module has a ∇-filtration Now Theorem 9.25 clearly implies that Rm,t as a Sm,t so its dual Sm,t has a Δ-filtration. opp (∇ (V ∗ )) = Furthermore, by duality, EndSm,t (Δ(λ)) = F if and only if EndSm,t λ F and this we know by Remark 9.35 and Theorem 9.34. 

10. RATIONAL GL(m)-MODULES

75

10. Rational GL(m)-modules 10.1. ∇-filtration. Using Definition 9.14 we now extend the notion of ∇filtration to any rational GL(m)-module. Definition 10.1. An arbitrary homogeneous (Proposition 2.19 2.) finite dimensional GL(m)-module M has a ∇-filtration if there is a sequence of submodules 0 = M0 ⊂ M1 ⊂ · · · ⊂ Mr = M such that Mj /Mj−1  ∇λ(j) for each j = 1, . . . , r and a suitable dominant λ(j) . In other words an arbitrary finite dimensional GL(m)-module M has a ∇filtration if there is some positive k so that detk ⊗M is a polynomial representation, which then decomposes into its homogeneous components and each such component of some degree i has a ∇-filtration as module on the corresponding Schur algebra. 10.1.1. Invariants. In particular this analysis applies to the study of the invariants M GL(m) which may appear only in the homogeneous of degree 0 of M . So let us assume that M is homogeneous of degree 0 with a ∇-filtration so that detk ⊗M is a polynomial representation of degree km with a ∇-filtration as module on the corresponding Schur algebra. Then the space M GL(m) corresponds in detk ⊗M to the contribution in the ∇-filtration of the bottom part relative to detk = ∇km . h

k

Proposition 10.2. Given an exact sequence 0 → N → M → L → 0 of GL(m)-modules with a ∇ filtration the sequence h

k

0 → N GL(m) → M GL(m) → LGL(m) → 0 is exact. Proof. Since taking a given homogeneous component is an exact functor, we deduce the statement from Corollary 6.14.  10.1.2. Characters. A finite dimensional GL(m, F ) module has a basis of weight vectors vχ , χ = (χ1 , . . . , χm ) ∈ Zm for the maximal torus of diagonal matrices T := {(x1 , . . . , xm ), xi ∈ F ∗ }: (x1 , . . . , xm )vχ = xχ vχ , xχ =

m 

xχi i .

i=1

Thus M = ⊕χ∈Zn Mχ with Mχ the subspace of vectors of weight χ. This allows us to define the character of M to be χ(M ) := χ dim Mχ xχ . It is easily seen to be a symmetric function since the normalizer of D in GL(m, F ) is the semidirect product of D with the symmetric group of permutation matrices Sm . For a Young diagram λ, the Schur module ∇λ is defined over any field F and is irreducible in characteristic 0. Also, since ∇λ has the basis of semistandard tableaux of shape λ and these are weight vectors, it follows that its character is independent from F and is given by the Schur symmetric function sλ (x1 , . . . , xm ) [41], introduced in 3.1.1. In general if k is a non negative integer such that km + λ is a partition, the character of ∇λ is the function sλ (x) = (x1 · · · xm )−k skm +λ . For an exact sequence 0 → N → M → P → 0 one has for every χ an exact sequence 0 → N χ → M χ → P χ → 0. Hence for the characters we have χ(M ) = χ(N ) + χ(P ).

76

III. THE SCHUR ALGEBRA

Since the Schur symmetric functions form an integral basis of the space of symmetric functions we have: Corollary 10.3. If M is a rational module with a ∇-filtration the multiplicity of a term ∇λ equals the coefficient of the symmetric function sλ (x) in the character of M . According to the definition of Remark 9.13 one may define a formal module structure over GL(m) for a finite rank free abelian group M and assume that it has a ∇-filtration. One easily verifies that, for every infinite field F one obtains a GL(m, F ) module M ⊗Z F with a ∇-filtration. In particular  we have a module M ⊗ Q = MQ with the same character as M . Then MQ = λ mλ ∇λ and we can say that the multiplicity mλ is also the multiplicity of ∇λ in any ∇ filtration of MF for every field F . In particular Corollary 10.4. The dimension of the space of invariants of MF for every infinite field F equals the dimension of the space of invariants of MQ . 10.2. Branching and truncation. Take a GL(m) module M . Consider the subgroup GL(m − 1) × GL(1) ⊂ GL(m). Recall that every character χ of GL(1) is of the form χ(t) = ti , i ∈ Z. Define T r i (M ) = T r χ M := {v ∈ M | t(v) = χ(t)v = ti v, ∀t ∈ GL(1)}. T r χ (M ) is a GL(m − 1) module which is a direct factor of M considered as a GL(m − 1) module. It is called the truncation of M by the character χ. If M is homogeneous of degree d (Proposition 2.19 2.) then T r i (M ) is a homogeneous representation of GL(m − 1) of degree d − i. Every rational representation M of GL(m) decomposes as M = ⊕i∈Z T r i M. Clearly Proposition 10.5. For every character χ, the functor M → T r χ M is exact. We need a simple definition Definition 10.6. Given a Young diagram λ its rim is the set of last (right) boxes of each row. r r r Observe that if T is a semistandard tableau with entries from 1, . . . , m of shape λ then m appears only in boxes in the rim. Moreover removing these boxes one has still a diagram and a semistandard tableau. In particular we may apply this discussion to the Schur module ∇λ (F m ) with λ a Young diagram, λ  d. Fix i, and set ∇iλ (F m ) := T r i (∇λ (F m )). Then ∇iλ (F m ) has as basis all the semi standard tableaux of shape λ with m appearing exactly i times. Consider the set Dλi of Young diagrams μ  d − i contained in λ, with at most m − 1 columns and obtained from λ by removing i boxes from the rim. By the previous remark, given a semistandard tableau T ∈ ∇iλ (F m ), if we remove all its i boxes containing m we get a semistandard tableau of shape μ(T ) ∈ Dλi .

10. RATIONAL GL(m)-MODULES

77

For μ ∈ Dλi we set ∇μλ (F m ) ⊂ ∇iλ (F m ) equal to the span of the semistandard  m ν m tableaux T ∈ ∇iλ (F m ) such that μ(T ) ≤ μ and ∇ 0 the left hand side is 0 so the statement is trivially true. Otherwise, by Lemma 11.16 and using the previous remark this follows from Proposition 11.14. 

Part IV

Matrix functions and invariants

88

IV. MATRIX FUNCTIONS AND INVARIANTS

This part is devoted to the proof of Theorem 1.10. 12. A reduction for invariants of several matrices We are now ready to give a first description of the invariants of matrices, that is Theorem 12.2. This is the main reason to have developed all the formalism of Part III of which this Theorem is a direct application. We are going to work over an infinite field F and as usual fix a basis e1 , . . . , em of the m-dimensional space V . We have seen in Theorem 1.4 that for a single matrix ξ = (xi,j ) , any invariant in S[ξ] = F [xi,j ] is a polynomial in the coefficients σh (ξ) , h = 1, . . . , m, of the characteristic polynomial of the generic ξ, that is the functions given, for A ∈ Mm (F ) by σh (A) = tr(∧h A) where ∧h A is the linear transformation induced by A  in h V . In what follows we choose a total order on Young diagrams refining the dominance order. For d ≥ 0, take the homogeneous component S d [ξ] of S[ξ]. We know by Theorem 8.25 that S d [ξ] has as basis the double semistandard tableaux whose shapes are Young diagrams with d boxes, λ  d, filled with positive integers less than or ˇ = k1 ≥ . . . ≥ km ≥ 0, be the row shape. equal than m. Let λ Recall that by Proposition 9.5 and Corollary 9.6, the space Tλ spanned by all bitableaux of shape λ  d, is a G := GL(V ) × GL(V )-submodule (here we use the fact that V ∗ is itself a GL(V )-module) and it is the image of the equivariant map,  ki  ki ∗  V, φi ∈ V , which on a decomposable element θ = i φi ⊗ ui , with ui ∈ on ξ ∈ End(V ) is: : πλ,V ˇ

ˇ λ 



V ⊗

ˇ λ 

V → S [ξ], πλ,V ˇ ( d



φi ⊗ ui )(ξ) :=



i

φi |

ki 

ξ(ui )

i

Equivalently, identifying ˇ λ 

V ⊗

ˇ λ 

V ∗ = End(

ˇ λ 

V)=



End(

ki 

V)

i

(51)

πλ,V ˇ (θ)(ξ) = tr(θ ◦

ˇ λ 

ξ) = tr(θ ◦

k1 

ξ⊗

k2 

ξ ⊗ ...⊗

km 

ξ) .

ˇ = h, and π ˇ (id)(ξ) = tr(h ξ) = σh (ξ). Notice that if λ = 1h , λ λ,V d By Definition 9.18,  the standard filtration of the algebra S [ξ] consists of the subspaces Mλ (V ) = μ≤λ Tμ , which, if no confusion arises, we shall simply denote by Mλ , and, by Corollary 9.20, if λ is the predecessor of λ in our total order (if λ is the minimum element Mλ = {0}), we know that Mλ /Mλ  ∇λ (V ∗ ) ⊗ ∇λ (V ). By Proposition 6.15 this implies that, as a G-module, Mλ has a ∇-filtration and hence by Theorem 11.1, restricting to GL(V ) embedded diagonally in G = GL(V )× GL(V ), also a ∇ filtration as a GL(V )-module. If we denote by Tst λ ⊂ Tλ the space spanned by double semi standard tableaux of shape λ, we have that Mλ = Mλ ⊕Tst λ.

12. A REDUCTION FOR INVARIANTS OF SEVERAL MATRICES

89

The map qλ : Mλ → Mλ /Mλ = ∇λ (V ∗ ) ⊗ ∇λ (V ) is surjective and πλ,V induces an ˇ exact sequence 0→

(52)

−1 πλ,V ˇ

(Mλ )



ˇ λ 



V ⊗

ˇ λ 

qλ ◦πλ

V −→ ∇λ (V ∗ ) ⊗ ∇λ (V ) → 0.

−1 Then, Lemma 6.12 implies that the module πλ,V ˇ (Mλ ) has a ∇-filtration as a Gmodule. Again, restricting to GL(V ) and using Theorem 11.1, we deduce that (52) is an exact sequence of GL(V )-modules with a ∇-filtration. By Proposition 10.2 also the sequence −1 GL(V ) 0 → πλ,V →( ˇ (Mλ )

ˇ λ 

V∗⊗

ˇ λ 

V )GL(V ) → (∇λ (V ∗ ) ⊗ ∇λ (V ))GL(V ) → 0

is exact. We now pass to the case of several matrices. Take the coordinate ring S[ξ1 , . . . , ξn ] of n copies of End(V ) = V ⊗ V ∗ . The group Gn = GL(V )2n naturally acts on S[ξ1 , . . . , ξn ]. Let d := (d1 , . . . , dn ) be a multidegree and let us analyze the component S d := S d [ξ1 , . . . , ξn ] = S d1 [ξ1 ] ⊗ S d2 [ξ2 ] ⊗ . . . ⊗ S dn [ξn ] of multidegree d. We can apply the filtrations of each factor. Using the chosen order on partitions, refining the dominance order, we order the sequences Λ = (λ1 , . . . , λn ), λi  di lexicographically and we set, for Z = (ζ1 , . . . , ζn ),  (53) MΛ (V ) = Mζi (V ) ⊗ · · · ⊗ Mζn (V ) ⊂ S d1 [ξ1 ] ⊗ S d2 [ξ2 ] ⊗ . . . ⊗ S dn [ξn ]. Z≤Λ

The G -submodules MΛ (V ) form a filtration of S d such that, if for a given Λ, Λ

denotes its predecessor (if Λ is the minimum element MΛ (V ) = {0}), we have the exact sequence n



0 → MΛ (V ) → MΛ (V ) −→ ∇Λ V ∗ ⊗ ∇Λ V → 0 where ∇Λ V ∗ ⊗∇Λ V  ∇λ1 (V ∗ )⊗∇λ1 (V )⊗∇λ2 (V ∗ )⊗∇λ2 (V )⊗· · ·⊗∇λn (V ∗ )⊗∇λn (V ). Hence each MΛ (V ) has a ∇-filtration as a Gn -module. ˇ i , Γ = (γ1 , . . . , γn ) and Set γi := λ Γ 



V ⊗

Γ 

V :=

γ1 



V ⊗

γ1 

V ⊗

γ2 



V ⊗

γ2 

V ⊗ ...⊗

γn 



V ⊗

γn 

V.

Consider the morphism πΓ,V := ⊗i πγi ,V . Reasoning exactly as above we then get the exact sequence of Gn modules with a ∇-filtration (54)

0→

−1 πΓ,V

(M (V )) → Λ

Γ 



V ⊗

Γ 

V

qΛ ◦πΓ,V

−→ ∇Λ V ∗ ⊗ ∇Λ V → 0,

where πΓ,V maps to TΛ = ⊗Tλi ⊂ MΛ and by Formula 51:  (55) πΓ,V (θ (1) ⊗ . . . ⊗ θ (n) )(ξ1 , . . . , ξn ) = πγi ,V (θ (i) )(ξi )

90

IV. MATRIX FUNCTIONS AND INVARIANTS

  with θ (i) ∈ γi V ⊗ γi V ∗ , ξi ∈ End(V ), i = 1, . . . n. Again, if we restrict to GL(V ) embedded multidiagonally in Gn , Theorem 11.1 implies that MΛ has a ∇filtration as a GL(V )-module and (54) is a exact sequence of GL(V )-modules with a ∇-filtration. By Proposition 10.2 also the sequence (56) Γ Γ   qΛ ◦πΓ,V −1 0 → πΓ,V (MΛ )GL(V ) → ( V ∗ ⊗ V )GL(V ) −→ (∇Λ V ∗ ⊗ ∇Λ V )GL(V ) → 0 is exact. Definition 12.1. The subspace J(n) ⊂ S[ξ1 , . . . , ξn ]GL(V ) consists of those invariants which are linear combinations of elements of the form γ1 γn   (57) πΓ,V (θ) = tr(θ ◦ ξ1 ⊗ . . . ⊗ ξn ) Γ ∗ Γ GL(V ) V ⊗ V) for some Γ = (γ1 , . . . , γn ), γi  di . with θ ∈ ( Theorem 12.2. J(n) = S[ξ1 , . . . , ξn ]GL(V ) . Proof. Take a non zero invariant I ∈ S[ξ1 , . . . , ξn ]GL(V ) which we may assume to lie in a fixed multigraded component S d [ξ1 , . . . , ξn ]. Take the minimum Λ such that I ∈ MΛ . Γ ∗ Γ GL(V ) By the exact sequence (56), there is an element H ∈ ( V ⊗ V) (with ˇ such that qΛ ◦ πΓ,V (H) = qΛ (I). It follows that πΓ,V (H) − I ∈ (MΛ )GL(V ) Γ = Λ) and our claim follows by induction.  In order to prove Theorem 1.10 we have to show that all the elements of J(n) come from S.    For this we first describe the algebras EndGL(V ) ( Γ V ) = ( Γ V ∗ ⊗ Γ V )GL(V ) arriving at Formula (71). We then need a combinatorial study of this formula, performed in section 15 and its final outcome in section 15.2. For the moment let us observe, by restricting ξ to a diagonal matrix, that σi (ξ) = tr(∧i ξ). We are going to use this to give an interpretation Formula (21). Observe first that, by Formula (26), tr(σ −1 ◦ξ ⊗i ) = j tr(ξ j )hj depends only on the conjugacy class Cλ to which σ belongs, associated to the partition λ = 1h1 2h2 . . . ihi , of its cycle type. We can thus set Tλ (ξ) = tr(σ −1 ◦ ξ ⊗h ) for σ ∈ Cλ . Using the fact i that the the orbit Cλ has i!/zλ elements, where zλ = j=1 hj !j hj is the cardinality of its centralizer, Formula (21) gives  1 1 1  σi (ξ) = λ Tλ (ξ) = λ cλ Tλ (ξ) = tr( σ σ ◦ ξ ⊗i ) = tr(∧i ξ). zλ i! i! λi

λi

σ∈Si

13. POLARIZATION AND SPECIALIZATION

91

13. Polarization and specialization We want to discuss polarization and specialization, which we have already introduced in Section 3.3.1, as general formal constructs which are valid in a wider contest. Usually these operations are defined on rings which in different ways are polynomial functions in some variables xi . They share the feature that linear substitutions of the variables xi induce polynomial actions on the entire ring. In order to have a general approach we start from a polynomial representation M , homogeneous of degree d, of GL(N, F ) where N = ∞ is also allowed (as a limit). Thus the action of GL(N ) extends to a multiplicative action of MN (F ) or to a module structure over the corresponding Schur algebra. Given various sets of commuting variables ui this will induce a multiplicative map of the polynomial ring MN (F )[ui ] on M [ui ]. Warning. An element a of MN (F ) or of MN (F )[ui ] induces a linear operator on M which we shall just denote by a, but sometimes it is necessary to make explicit the same operator as belonging to the Schur algebra and then we make it explicit as a⊗d . As usual denote by D the torus of diagonal matrices, t = (t1 , . . . , tN ). Let Xd N hi denote the set of characters χ : t → N i=1 ti of D with hi ≥ 0 and i=1 hi = d. Setting Mχ = {m ∈ M | (tm = χ(t)m}, we get a direct sum decomposition M = ⊕χ∈Xd Mχ . Examples are obtained by taking as M the homogeneous components of degree d of various algebras such as the polynomial ring, the free algebra, the trace algebra, the algebra S in N variables. In these cases the GL(N ) action is by algebra automorphisms, commuting in the relevant cases with trace or with the σ operations. In all these cases Mχ is the space of elements which are homogeneous of degree hi in the variables xi . We want to define the maps of specialization and polarization in this generality. Take a surjective map f : [1, . . . , N ] → [1, . . . , h] with Cj = f −1 (j) and cj = |Cj | and consider the subgroup

Notice that Sf =

h i=1

SCi

Sf := {σ ∈ SN | f ◦ σ = f }. h  i=1 Sci is a Young subgroup.

13.0.1. Specialization Qf . Specialization is given by applying the operator (58)

Qf =

N 

ef (i),i

i=1

acting on M . In all of the above examples, the operator Qf is just the specialization xi → xf (i) . We have that (

N  i=1

Therefore

ti ei,i )Qf =

N 

tf (i) ef (i),i .

i=1

hi Lemma 13.1. If m ∈  Mχ , χ = N i=1 ti we have that Qf m ∈ Mf (χ) , with N h i h h −1 i f (χ) = i=1 tf (i) = j=1 tj i∈f (j) .

92

IV. MATRIX FUNCTIONS AND INVARIANTS

13.0.2. Polarization Pf . Given h ≤ N, h ∈ N denote by Dh the subgroup of D where ti = 1, ∀i ≤ h and M (h) the subspace of elements of M invariant under Dh . In the usual polynomial case it is the subring depending on the first h variables. Lemma 13.2. M (h) is the direct sum of the Mχ , χ = hi=1 thi i ,  i

hi ≥ 0, hi = d. M (h) is a polynomial representation of GL(h) homogeneous of degree

d. Proof. Clearly M (h) is the direct sum of the Mχ invariant under Dh and this is the given condition. Since GL(h) commutes with Dh the second statement follows.  We can now define Pf the polarization operators. For simplicity assume N < ∞ and take commuting variables z1 , . . . , zN . Consider the map πf : M (h) → M [z1 , . . . zN ] given by the operator πf =

N 

zj ej,f (j) .

j=1

In the case of polynomials  in variables xj the operator πf is just the substitution of the h variables xi → j | i=f (j) zj xj . Since M is a polynomial representation of degree d the map πf maps M (h) to polynomials in the zi s homogeneous of degree d.  Let us now fix a sequence s := {s1 , . . . , sh }, si ∈ N, i si = d. s

Definition 13.3. For a ∈ M (h), we define Pf (a) to be the coefficient of the monomial di=1 zisi of πf (a). d In particular we set Pf (a) = Pf1d (a), the coefficient of the monomial i=1 zi of πf (a). s

We call the map Pf a map of partial polarization and call the map Pf the map of full polarization with respect to f . We have that d d   ( tj ej,j )πf = tj zj ej,f (j) . j=1

j=1

Therefore s

Lemma 13.4. The operator Pf maps M (h) to the subspace Mχs , χs =

N

si i=1 ti .

The space of eigenvalue χ1d := di=1 ti is also called the space of multilinear elements. This space is stable under the action of the symmetric group of permutation matrices Sd ⊂ GL(d) ⊂ GL(N ). Assume N = d, we then claim that the image of Pf is contained in the subspace  of elements invariant under Sf . In fact if σ = di=1 eσ(i),i ∈ Sd we have σ◦

d  j=1

zj ej,f (j) =

d  j=1

zj eσ(j),f (j)

13. POLARIZATION AND SPECIALIZATION

93

so if σ ∈ Sf that is f (j) = f (σ(j)) we have σ◦ and since

d

zj ej,f (j) =

j=1

i=1 zi

Qf ◦ πf =

d 

=

d 

d

i=1 zσ −1 (i)

i=1

(59)

zj eσ(j),f (σ(j)) =

j=1

ef (i),i ◦

So if a ∈ Mχc , χc =

d 

d 

ci i=1 ti ,

Qf ◦ πf (a) =

h 

(

zσ−1 (j) ej,f (j)

j=1

the claim follows. Now

zj ej,f (j) =

j=1

h

d 

d 

zj ef (j),f (j) =

h 

(



zj )ei,i .

i=1 j | f (j)=i

j=1

we get 

zj )ci a and Qf ◦ Pf (a) =

i=1 j | f (j)=i

h 

ci !a.

i=1

Remark 13.5. As was previously mentioned, because of this formula the op erator Qf / hi=1 ci ! is sometimes called restitution operator. Let us now try to understand what happens when we iterate the procedures of polarization and specialization. Assume that the map f : [1, . . . , d] → [1, . . . , h] is the composition of the two maps: m : [1, . . . , d] −→ [1, . . . , k], g : [1, . . . , k] −→ [1, . . . , h],  di := |g −1 (i)|, ci := |f −1 (i)| mj := |m−1 (j)|, ci = mj . j | g(j)=i

In this situation we will say that m or rather the partition induced by m is a refinement of f . We clearly have Qf = Qg ◦ Qm . Let us see what happens for the full polarization Pf . Let us write πf = d d j=1 zj ej,f (j) = j=1 zj ej,g◦m(j) and, using different variables u and wj ,  ∈   [1, . . . , d], j ∈ [1, . . . , k], πm = d=1 u e,m() πg = kj=1 wj ej,g(j) . We have    u e,m() wj ej,g(j) = wm() u e,g◦m() . πm ◦ πg = ∈[1,...,d]

j∈[1,...,k]

∈[1,...,d]

So the composition of πm ◦ πg can also be interpreted as πf composed with the d variable substitution z → wm() u . The coefficient Pf (a) of the product =1 z d in πf (a) is thus the coefficient of =1 wm() u in πm ◦ πg (a). Finally this is the d d k m coefficient of =1 u in πm applied to the coefficient of =1 wm() = j=1 wj j in πg (a). That is: Pm ◦ Pgm = Pf : Mc → Mm → M1d .

(60)

Let us now consider d d d    uk ek,k ) = zj ej,f (j) ◦ ui ef (i),i = Hf = πf ◦ Qf ◦ ( j=1

k=1

 j,k | f (j)=f (k)

i=1

Let us now take a multilinear element a ∈ Mχ1d . We get Hf (a) = (

d  i=1

ui )πf ◦ Qf (a) =

d  i=1

ui

 j,k | f (j)=f (k)

zj ej,k (a).

zj uk ej,k .

94

IV. MATRIX FUNCTIONS AND INVARIANTS

In order to expand this as polynomial in z we need to work in the Schur algebra Sd,d . d d s,t We define Hf to be the coefficient of the monomial i=1 zisi i=1 utii of Hf⊗d . s,t We have by definition Hf⊗d ∈ Sd,d [z, u], so Hf is an element in the Schur algebra. Thus d  s,1d πf ◦ Qf (a) = ( zisi )Hf (a) s

i=1

and

d

d

Pf ◦ Qf (a) = Hf1 ,1 (a). Recall that a permutation σ ∈ Sd is represented in the Schur algebra by the operator  eσ(k1 ),k1 ⊗ . . . ⊗ eσ(kd ),kd σ ⊗d = (k1 ,...,kd )

so its restriction to Mχ1d is given by  eσ(k1 ),k1 ⊗ . . . ⊗ eσ(kd ),kd . σ ⊗d = (k1 ,...,kd )∈Sd

On the other hand d

Hf1

,1d

=



ei1 ,k1 ⊗ . . . ⊗ eid ,kd

(i1 ,...,id ); (k1 ,...,kd )

with (i1 , . . . , id ), (k1 , . . . , kd ) ∈ Sd and f (ij ) = f (kj ). This means that in Sd , (i1 , . . . , id ) = τ (k1 , . . . , kd ) for a unique τ ∈ Sf . It follows that for a multilinear, one has  τ )a. (61) Pf ◦ Qf (a) = ( τ ∈Sf

In particular from (59) and (61) we deduce Proposition 13.6. (1) The restriction of Qf ◦ Pf to the space Mχc is h given by multiplication by i=1 ci !. S (2) The restriction of Pf ◦ Qf to the space Mχ1fd is given by multiplication by h i=1 ci !. s

More generally the reader can easily verify that the restriction of Qf ◦ Pf to

ci . the space Mχc is given by multiplication by hi=1 sj ,...,s j 1

ci

14. EXTERIOR PRODUCTS

95

14. Exterior products 14.1. The Hecke algebras. Let us start with a Definition. Definition 14.1. A composition d  N of an integer N is a map d : N → Z≥0  with finite support such that j d(j) = N . A partition is then a non increasing composition. We are often going to write dj for d(j). To a composition d  N we associate the surjective map f : [1, . . . , N ] → i−1 i [1, . . . h] defined by f (j) = i for s=1 ds < j ≤ s=1 ds . On the other hand from a surjective map f : [1, . . . , N ] → [1, . . . , h] we deduce the composition dj = |f −1 (j)|. Clearly in this situation there is a permutation σ ∈ SN with the property that f ◦ σ is the surjection associated to the composition d. In view of this all the results of this section can be easily generalized to the case of general surjections. h Definition 14.2. The subgroup Sf  i=1 Sdi of permutations in SN which commute with f is called the Young subgroup associated to f . Let V be a finite dimensional free module over a commutative ring A. Fix a composition d  N and let f : [1, . . . , N ] → [1, . . . , h] be the corresponding surjection, we set d dh d1    V = V ⊗ . . . ⊗ V. Consider the almost idempotent: Af :=

(62)



σ σ,

A2f = |Sf |Af .

σ∈Sf

By remark 9.8 and using Formula (44) we can identify mapping (63)

d

V with Af V ⊗N by

id1 (v1 ∧ . . . ∧ vd1 ) ⊗ . . . ⊗ idh (vN −dh +1 ∧ . . . ∧ vN ) = Af (v1 ⊗ . . . ⊗ vN )

When A = Q we can take the idempotent 1  Af ed := ∈ EndGL(V ) V ⊗N , σ σ = |Sf | |Sf |

e2d = ed .

σ∈Sf

d We have also V = ed V ⊗N . Given two compositions d = (d1 , . . . , dh ), c = (c1 , . . . , cr ) of N , denote with f : [1, . . . , N ] → [1, . . . , h] and k : [1, . . . , N ] → [1, . . . , r] the associated surjections with the corresponding subgroups Sf  Sd1 × Sd2 × . . . × Sdh , Sk  Sc1 × Sc2 × . . . × Scr ⊂ SN . Let ed , ec be the corresponding antisymmetrizers. Assuming that F = Q we have: Proposition 14.3. (64)

homGL(V ) [ed V ⊗N , ec V ⊗N ] = ec EndGL(V ) V ⊗N gd.

If dim V = m ≥ N we have homGL(V ) [ed V ⊗N , ec V ⊗N ] = Ac,d (Q) where (65)

Ac,d (Q) := {x ∈ Q[SN ] | σx = σ x, ∀σ ∈ Sk , σ x = xσ, ∀σ ∈ Sf }.

96

IV. MATRIX FUNCTIONS AND INVARIANTS

Proof. We decompose V ⊗N = ec V ⊗N ⊕ (1 − ec )V ⊗N = ed V ⊗N ⊕ (1 − ed )V ⊗N both GL(V ) stable decompositions. Then we decompose EndGL(V ) V ⊗N as block 2 × 2 matrices and identify the block homGL(V ) [ed V ⊗N , ec V ⊗N ] with the linear operators from V ⊗N to ec V ⊗N which are 0 on (1 − ed )V ⊗N . If dim V ≥ N we have that EndGL(V ) V ⊗N = Q[SN ] so we have that Formula (64) implies Formula (65).   Corollary 14.4. Ac,d (Q) has as basis the elements uD := |S1f | σ∈D σ σ indexed by double cosets D = Sk τ Sf . 

Proof. This follows immediately from Formula (65).

d V which We can give an explicit formula for the action of an element uD on shows that it is in fact defined over Z and thus over any commutative ring A. Given a double coset D, choose a left transversal L(D) that is a set L(D) such  that D = ˙ σ∈L(D) σSf is a disjoint union (and a right transversal R(D) such that  D= ˙ Sk σ ). Set σ∈L(D)

(66)



θD =

σ σ,

θ¯D =

σ∈L(D)

We have that (67)





σ σ.

σ∈R(D)

σ σ = θD Af = Ak θ¯D , so that uD = θD ed .

σ∈D

Since ed is the identity on this it follows:

d

V we have that uD coincides on

14.5. The element uD induces, between  c Proposition V = Ak V ⊗N , the map θD = σ∈L(D) σ σ.

θD

d

d

V with θD . From

V = Af V ⊗N and

Corollary 14.6. The operators θD are defined over any commutative ring A. If V is a vector space over any field F and m = dim V ≥ N the linear operators d c are linearly independent and form a basis of homGL(V ) ( V, V ).

Proof. The first statement is clear. As for the second, let us remark that, for any double coset D, and N linearly independent vectors e1 , . . . , eN of V we have  σ σAf (e1 ⊗ e2 ⊗ . . . ⊗ eN ). (68) θD (Af (e1 ⊗ e2 ⊗ . . . ⊗ eN ) = σ∈L(D)

The fact that the elements θD are linearly independent follows from the fact that they are sums, with signs, over disjoint sets of the linearly independent vectors σ(e1 ⊗e2 ⊗. . .⊗eN ), σ ∈ SN . The fact that they form a basis is then a consequence of d c Corollary 10.4, since homGL(V ) ( V, V ) has a ∇-filtration hence the dimension d c V ) is independent of the characteristic and over Q we have of homGL(V ) ( V, seen (Corollary 14.4) that the given elements form a basis.  We now define Definition 14.7. Ac,d := ⊕D ZuD , the integral form of Ac,d (Q).

14. EXTERIOR PRODUCTS

97

For every commutative ring A we set Ac,d (A) := Ac,d ⊗Z A. Let F be an infinite field. Consider the mapping Φ : Ac,d (F ) → homGL(V ) (

(69)

d 

V,

c 

V ),

Φ(uD ) := θD .

Proposition 14.8. (1) If dim V ≥ N the map Φ is an isomorphism. (2) If c = d, Φ is also an isomorphism of algebras and in this case we set Ad (F ) := Ad,d (F ). (3) For every vector space V over the infinite field F the map Φ is surjective. Proof. (1) and (2) follow from Corollary 14.6. The case dim V < N follows, using truncation, from Proposition 11.14.  Definition 14.9. Consider the map ρ : Ac,d (F ) ◦ Ad,c (F ) → Ac (F ) given by composition. The span of the elements of Ac (F ) which are in the image of ρ will d be denoted by Ac (F ). d

Proposition 14.10. The space Ac (F ) is formed by linear combinations of the elements    1 (70) gD1 ,D2 , gD1 ,D2 := σ1 σ2 τ σ1 τ σ2−1 , |Sk | τ ∈Sf

σ1 ∈L(D1 ) σ2 ∈L(D2 )

with Di = Sk τi Sf double cosets and L(Di ) chosen left and right transversals. Proof. As soon as we take dimF V ≥ N, Corollary 14.6 implies that we can identify ρ : Ac,d (F ) ◦ Ad,c (F ) → Ac (F ) to the composition homGL(V ) (

d 

V,

c 

V ) ◦ homGL(V ) (

c 

V,

d 

V ) → EndGL(V ) (

d 

V ).

Thus the elements which factor are linear combinations of products uD1 ◦ uD−1 for two double cosets D1 = Sk τ1 Sf , D2−1 = Sk τ2 Sf . Write    1 1 uD 1 = σ1 σ1 ( τ τ ), uD−1 = 2 |Sf | |Sk | τ ∈Sf

σ1 ∈L(D1 )

Recall that (



σ2 ∈L(D2 )

τ τ )2 = |Sf |(

τ ∈Sf



2

σ2 (



τ τ )σ2−1 .

τ ∈Sf

τ τ ).

τ ∈Sf



So, multiplying we deduce the desired relation.

Take F = Q. We want to compute, given the generic matrices ξ1 , . . . , ξh ∈ End(V ), the function tr(θD ◦ ∧d1 ξ1 ⊗ . . . ⊗ ∧dh ξh ). We have Proposition 14.11. Let D = Sf τ Sf be a double coset. Then 1  σ tr(σ ◦ ξf (1) ⊗ . . . ⊗ ξf (N ) ) tr(θD ◦ ∧d1 ξ1 ⊗ . . . ⊗ ∧dh ξh ) = |Sf | (71) σ∈D

= tr(uD ◦ ξf (1) ⊗ . . . ⊗ ξf (N ) ). Proof. We clearly have the decomposition V ⊗N = ed V ⊗N ⊕ (1 − ed )V ⊗N =

d 

V ⊕ (1 − ed )V ⊗N .

98

IV. MATRIX FUNCTIONS AND INVARIANTS

We know, from Formula (67), that uD = uD ed , that is uD (1 − ed ) = 0. By definition the operator ∧d1 ξ1 ⊗ . . . ⊗ ∧dh ξh is the restriction to ed V ⊗N of the map ξf (1) ⊗ . . . ⊗ ξf (N ) (which preserves the decomposition). Hence tr(θD ◦ ∧d1 ξ1 ⊗ . . . ⊗ ∧dh ξh ) = tr(θD ◦ ed ◦ ξf (1) ⊗ . . . ⊗ ξf (N ) ) and the claim follows from the fact that θD ◦ ed = uD .



Observe that, if V = QN and the ξi are evaluated into matrices over Z, the left hand side of Formula (71) is the trace of an operator on a free Z module so that its trace is a polynomial in the entries of the ξi with integer coefficients. On the other hand the right hand side has a priori rational coefficients. We will see in §15.2 using Formula (80), that in fact the denominator disappears.

15. MATRIX FUNCTIONS AND INVARIANTS

99

15. Matrix functions and invariants 15.1. Graded subspaces. As before, let X = {x1 , . . . , xi , . . .} be a finite or countable alphabet which we are going to consider as infinite matrix variables. In section 4.2.2, for any commutative ring A, we introduced the algebra SA = SA X which can be viewed either as a symbolic algebra or, at least if A is an infinite domain, as functions in matrix variables in M∞ (A). In particular we have set SZ = S. Our final goal, which will be achieved in 15.2, is to understand Formula (71) as evaluation of an element of S. Recall that by Corollary 4.23, S equals the polynomial ring Z[σi (p)] in the variables σi (p), p ∈ W0 a primitive monomial in the alphabet X (up to cyclic equivalence). This induces a multigrading on the algebra S defined as follows. Every primitive monomial (up to equivalence) p ∈ W0 has a multidegree, which we denote by deg p. Then, since for each i, σi is a polynomial map of degree i, it is natural to give σi (p) degree i (p) and multidegree i deg p a composition of i (p) (Def. 14.1). We denote by S(c) the multigraded component of S of multidegree c. In this section we are going to show various properties of these components. For a given element p ∈ W0 we can consider the subring S(p) ⊂ S of polynomials in the variables σi (p). This has a basis indexed by partitions as follows. For a λ(1) λ(2) partition 2 . . . kλ(k) written in the form of Formula (37), of the integer  λ=1 |λ| = j λ(j)j and to p ∈ W0 we take the element  σj (p)λ(j) ∈ S(p) (by convention σ0 (p) = 1, ∀p). σλ (p) := j

Notice that (σλ (p))= has multidegree |λ|deg(p). Clearly Proposition 15.1. The elements σλ (p), as λruns over all partitions, are a homogeneous basis of S(p) (over Z) moreover S = p∈W0 S(p). By Proposition 4.21, SQ (p) is also a polynomial ring in the variables σ1 (pk ), so an alternative homogeneous basis for SQ (p) (over Q) can be given by the elements  (72) tλ (p) := σ1 (pj )λ(j) ∈ SQ (p) j

The change of basis is given by Formula (21). Let us now fix a multidegree c (a composition). Denote by P the set of partitions, including the empty partition. Definition 15.2. We set (73)



Θc := {μ : W0 → P | c =

|μ(p)|deg p}.

p∈W0

For any μ ∈ Θc we can then consider the elements    Dμ := σμ(p) (p) = σj (p)μ(p)(j) ∈ S(c),

(74)

Tμ :=

p∈W0

p∈W0 j



 

p∈W0

tμ(p) (p) =

σ1 (pj )μ(p)(j) ∈ S(c).

p∈W0 j

The following is a corollary of Proposition 15.1:

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IV. MATRIX FUNCTIONS AND INVARIANTS

Proposition 15.3. (1) The elements Dμ with μ ∈ Θc form a basis of S(c). (2) The elements Tμ form a basis of SQ (c) with change of basis given by Formula (21). We start by studying the multilinear part of S of degree N , that is the part of S of multidegree c := 1N given by 1, 1 ≤ i ≤ N ci = 0, i > N . Lemma 15.4. We have a canonical bijection σ → Tμσ between the symmetric group SN and the set of Tμ with μ ∈ Θ1N given by (75)

σ → Tμσ = Tσ := tr(σ −1 ◦ x1 ⊗ x2 ⊗ . . . ⊗ xN ),

xi ∈ X.

Proof. Given a permutation σ ∈ SN , let us write σ = c1 . . . ct , a product of disjoint cycles cj = (r1 , . . . rhj ). The sequence r1 , . . . rhj is uniquely defined up to cyclic equivalence, and xr1 xr2 . . . xrhj is clearly primitive since it is a product of distinct variables, so we may assume that r1 = min(r1 , . . . rhj ) and thus the monomial pj = xr1 xr2 . . . xrhj ∈ W0 . For σ ∈ SN we define μσ : W0 → P and Tσ ∈ S by (1), if p = pj for some j μσ (p) = . ∅, otherwise. The fact that μσ ∈ Θ1N and that the map σ → Tμσ gives a bijection is then clear. Furthermore Formula (75) is equivalent to Formula (26).  Remark 15.5. Notice that for μσ ∈ Θ1N , the elements Dσ := Dμσ and Tσ = Tμσ coincide and give a basis for S(1N ). Extending  the map T to a map T : Z[SN ] → S(1N ) defined by  by linearity setting T ( σ aσ σ) := σ aσ Tσ , we identify S(1N ) with the group ring Z[SN ]. When analyzing S(c) for a general composition, we can, up to permuting the variables xi , assume that the composition c = (c1 , c2 , . . . ch ) with ci > 0 for each i ≤ h is associated to a surjective map f : [1, . . . , N ] → [1, . . . , h] with cj = |f −1 (j)|. To f it is also associated the Young subgroup Sf of Definition 14.2. Remark 15.6. The notion of Young subgroup is just associated to any partition of [1, . . . , N ] or to any map f : [1, . . . , N ] → T , T a set. Given two maps fi : [1, . . . , N ] → Ti , i = 1, 2 the subgroup Sf1 ∩ Sf2 is also a Young subgroup Sf1 ×f2 associated to the product map to T1 × T2 . We want do apply to S the operations, Qf : S(x1 , . . . , xN ) → S(x1 , . . . , xh ), Pf : S(x1 , . . . , xh ) → S(x1 , . . . , xN ) of specialization and polarization introduced in §13: Here by S(x1 , . . . , xj ) we denote the part of S which depends only on the variables (x1 , . . . , xj ). Similarly denote W0 (x1 , . . . , xj ) the elements of W0 in the variables x1 , . . . , xj .

15. MATRIX FUNCTIONS AND INVARIANTS

101

15.1.1. Specialization Qf . The map Qf of specialization can be defined from the formal substitutional rules of S. We have that f induces a map S → S, Qf : xi → xf (i) . Warning. The ring S(x1 , . . . , xN ) is the polynomial ring over the elements σi (q) where q = xm1 · · · xms is a primitive monomial in the variables x1 , . . . , xN . However Qf (q) = pr with p primitive so, if r > 1, Qf (σi (q)) = σi (Qf (q)) is not one of the generators of S(x1 , . . . , xh ) but it is expressed as a polynomial in the generators σj (p). c

Write Qf (q) := xf (m1 ) · · · xf (ms ) ∼ pr with p ∈ W0 (x1 , . . . , xN ). This allows us to define the map: Πf : W0 (x1 , . . . , xN ) → W0 (x1 , . . . , xN ) × N

(76)

by

Πf (q) := (p, r).

Let us now take a permutation σ ∈ SN and decompose σ as a product of disjoint cycles σ = c1 · · · cm . It follows that: Definition 15.7. To σ and f we can associate the map fσ ∈ Θc defined by setting fσ (p)(j) equal to the number of cycles ct decomposing σ, such that Πf (ct ) := (p, j). Considering x1 , . . . , xN as matrix variables, we set, using Formulas (72), (75) and (74)  tfσ (p) . (77) Tfσ := Qf (Tσ ) = tr(σ ◦ xf (1) ⊗ . . . ⊗ xf (N ) ) = p∈W0

Example 15.8. N = 4, f : 1, 2, 3, 4 → 1, 1, 2, 2. If σ = (1, 3)(2, 4) then fσ (p)(j) = 0 unless p = x1 x2 , then fσ (x1 x2 )(1) = 2 and Tfσ = t(x1 x2 )2 . If σ = (1, 3)(2)(4) the fσ (p)(j) = 0 unless p = x1 x2 , j = 1 or p = x1 , j = 1; p = x2 , j = 1. Tfσ = t(x1 x2 )t(x1 )t(x2 ). If σ = (1, 2)(3, 4) then fσ (x1 )(2) = fσ (x2 )(2) = 1 and Tfσ = t(x21 )t(x22 ). If σ = (1, 3, 2, 4) then fσ (x1 x2 )(2) = 1 and Tfσ = t((x1 x2 )2 ). If σ = (1, 2, 3, 4) then fσ (x21 x22 )(1) = 1 and Tfσ = t(x21 x22 ). Observe now a general fact, let f : [1, . . . N ] → [1, . . . , h] be a surjective map.   Definition 15.9. Let A = σ∈SN aσ σ, B = σ∈SN bσ σ.  We say that B ≡f A if B = σ∈SN aσ τσ στσ−1 , for some choice of τσ ∈ Sf . Then Lemma 15.10. If A ≡f B we have Qf (T (A)) = Qf (T (B)). Proof. It suffices to prove this for A = σ, B = τ στ −1 with τ ∈ Sf . The cycle decomposition of τ στ −1 is obtained from that of σ by substituting i with τ (i). Since τ ∈ Sf we have f (i) = f (τ (i)) so i, τ (i) have to be substituted with the same variable, the claim follows. 

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IV. MATRIX FUNCTIONS AND INVARIANTS

15.1.2. Properties of double cosets. As usual, we take a composition c  N associated to a surjection f : [1, . . . N ] → [1, . . . , h] and the subgroup Sf  hi=1 Sci of permutations in SN which commute with f . Let μ ∈ Θc , set Σμ := {σ ∈ SN | Qf (Tσ ) = Tμ }. Proposition 15.11. For every μ ∈ Θc , Σμ is an orbit under the conjugation action of Sf . Proof. The cycle structure of a permutation σ ∈ Σμ can be read from Formula (74), so it is determined by μ. It follows that any two permutations in Σμ are conjugate. Furthermore by pairing the cycles giving the same pi they can be  conjugated by a permutation commuting with f , hence lying in Sf . We are now going to determine for each μ ∈ Θc the cardinality of Σμ . Recall first a standard fact on permutations. Take a partition, as in Formula  (37), λ := 1λ(1) 2λ(2) . . . kλ(k)  |λ| = i≥0 i λ(i). Consider a permutation τ ∈ S|λ| whose cycle decomposition is given by λ, that is τ = c1 · · · cm with ct disjoint cycles such that for each i exactly λ(i) cycles among the ct ’s have length i. Then the cardinality of the centralizer of τ in S|λ| equals (cf. [48] Ch. 9, 4.1)   1 iλ(i) λ(i)! and = 1. (78) zλ = zλ i λm

We have Proposition 15.12. For any σ ∈ SN , the centralizer Zσ of σ in Sf has p∈W0 zfσ (p) elements. Proof. Write σ = c1 · · · cm , ct disjoint cycles. From the map f , using (76) we get a map {c1 , . . . , cm } → W0 × N. . , cm } → W0 . We can set for each Projecting to W0 we deduce a map δ : {c1 , . . p ∈ Im(δ), Cp = δ −1 (p) and it follows that σ = p σp with σp = ci ∈Cp ci . Since the cycles ct are disjoint, it is clear that we obtain a decomposition of [1, . . . , N ] into disjoint subsets Ipσ of cardinality |p||λσ (p)|, such that for each p, σp ∈ SIpσ . We first claim that, if τ ∈ Zσ , then τ preserves each of the sets Ipσ . In fact take a cycle c = (m1 , . . . , ms ) in Cp with corresponding monomial xf (m1 ) · · · xf (ms ) = pr , so the set {m1 , . . . , ms } ⊂ Ipσ . Since f ◦τ = f , the same monomial is associated to τ cτ −1 = (τ (m1 ), . . . , τ (ms )). Hence the set {τ (m1 ), . . . , τ (ms )} ⊂ Ipσ . Let Sp,f = SIpσ ∩ Sf = {τ ∈ SN | τ (Ipσ ) = Ipσ , f ◦ τ = f }. It follows that Zσ = p Zσp , Zσp being the centralizer of σp in Sp,f . So, it is enough to analyse the case in which only one p appears. In this case we have the usual argument. The permutation σ has cycle composition |p|λσ (p) and an element of Zσ permutes these cycles. We need to show first that every permutation of cycles of the same length is obtained. Take two cycles of equal length c = (i1 , . . . , i|p|k ), d = (j1 , . . . , j|p|k ) appearing in σ which hence both specialize to pk . By definition the permutation exchanging is with js for each s = 1, . . . , |p|k and fixing the remaining indices, lies in Sf and induces the transposition of c with d proving our claim. Thus we get a surjective homomorphism of

15. MATRIX FUNCTIONS AND INVARIANTS

103

Zσ onto Sλσ (p) whose kernel H is the product of the groups that cyclically permute each cycle (i1 , . . . , i|p|k ) but preserving f . For each such cycle c = (i1 , . . . , i|p|k ) a cyclic permutation cj preserves f if and only if j is a multiple of |p|, since p is primitive. Hence such permutations form a cyclic group with k elements. We  deduce that |Zσp | = zfσ (p) as desired. 15.1.3. Polarization. Proposition 15.13. For any μ ∈ Θc , we have Pf (Tμ ) = zμ

 σ∈Σμ

Tσ .

Proof. We have seen in Section 13 that the image of Pf is contained in the Sf of elements invariant under consubspace S(1N )Sf identified to the space Z[S N] jugation by Sf . Let us now write Pf (Tμ ) = σ∈SN aσ Tσ , aσ ∈ Z.  By the definition of Pf , aσ ≥ 0. By Proposition 13.6 we get σ∈SN aσ = |Sf | and aσ = 0 only if Qf (Tσ ) = Tμ that is σ ∈ Σμ . Since a = aσ does not depend on the choice of σ ∈ Σμ , we deduce |Sf | = a|Σμ |. By Proposition 15.12, a = zμ .  15.1.4. Young superclasses. Let us now consider the set  g(p)deg p = c}. Ξc := {g : W0 → N | p∈W0



In particular by composition

p∈W0

g(p)(p) = N . The map P → N defined by λ → |λ| induces, W0 → P → N,

μ : p → μ(p) → |μ(p)|

an obvious surjection s : Θc → Ξc (cf. Formula (73)). Notice that, given f and σ ∈ SN and p ∈ W0 we have |Ipσ | = (p)|fσ (p)| = (p)s(fσ )(p) (cf. Def. 15.7). We may thus introduce on SN the following equivalence relation depending on f f . We say that σ ∼ τ if s(fσ ) = s(fτ ), that is if, for all p, we have that |Ipσ | = |Ipτ |. Each equivalence class in SN is determined by a function g ∈ Ξc . Definition 15.14. The equivalence class SCg = {σ ∈ SN | s(fσ ) = g} will be called a Young superclass. Lemma 15.15. Each Young superclass has |Sf | elements. Proof. We have that the superclass associated to a function g ∈ Ξc is the union of the conjugacy classes, under Sf , associated to the functions μ ∈ Θc such that s(μ) = g. So their total number is    |Sf | 1 1 = |Sf | = |Sf | zμ p∈W0 zμ(p) p∈W0 zμ(p) μ | s(μ)=g

μ | s(μ)=g

= |Sf |



(



p∈W0 λg(p)

μ | μ(p)g(p)

1 ) = |Sf |, zλ

where the last equality follows from the identity (78).



Remark 15.16. For every function g ∈ Ξc we can choose the distinguished element μg ∈ Θc with s(μg ) = g given by μg (p)(j) = g(p)δ1,j , ∀p ∈ W0 . Definition 15.17. A simple degeneration of an element σ1 (pk ) with p primitive is a splitting σ1 (pk ) → σ1 (pi )σ1 (pj ), i + j = k. A simple degeneration of a product

104

IV. MATRIX FUNCTIONS AND INVARIANTS



σ1 (pki i ) is just the replacement in this product of a term σ1 (pki i ) with a simple degeneration. A degeneration is obtained by a sequence of simple degenerations. We say that a permutation τ is a degeneration of a permutation σ if the element Tμτ is a degeneration of Tμσ . i

Notice that an element has no proper degeneration if and only if it is of the form p∈W0 σ1 (p)g(p) , that is if the corresponding μ ∈ Θc is distinguished. Lemma 15.18. The Young superclass SCg of a function g ∈ Ξc is formed by all permutations σ such that μσ ∈ Θc degenerates to the distinguished element μg ∈ Θc . 

Proof. This is basically tautological.

Proposition 15.19. Each double coset of Sf σSf ⊂ SN is a union of superclasses. Proof. If two permutations τ, σ are conjugate under Sf they are clearly in the same double coset, so it is enough to show that if τ is a simple degeneration of a permutation σ then τ, σ are in the same double coset. Suppose that c = (i1 , . . . , i|p|k ) is a cycle of σ whose associated pair is (p, k) and we degenerate c = c c

with c = (i1 , . . . , i|p|s ), c

= (i|p|s+1 , . . . , i|p|k ), then c = c c

(i1 , i|p|s+1 ). Since f (i1 ) = f (i|p|s+1 ) the transposition (i1 , i|p|s+1 ) lies in  Sf and our claim follows. Before we discuss the main formula we need a little digression on signs. Let σ ∈ SN have a cycle decomposition σ = c1 · · · cm , given by a composition λ with N = |λ|. A cycle is an even permutation if and only if it is an odd cycle. It follows that the sign σ of σ equals λ := (−1)e(λ) where e(λ) = j λ(2j), denotes the number of even cycles in σ. If o(λ) denote the number of odd cycles in λ we have that o(λ) ∼ = |λ|, modulo 2. Given now a positive integer r, consider the function r λ defined by λ(j/r) if j ≡ 0 mod r r λ(j) = 0 otherwise. The function r λ is a composition of r|λ|. Clearly if r is odd we have e(r λ) = e(λ) so r λ = λ . Instead, if r is even, we have e(r λ) = e(λ) + o(λ). We deduce r λ = (−1)e(r λ) = (−1)e(λ) (−1)(r−1)o(λ) = λ (−1)(r−1)|λ|

In particular for a permutation σ in the superclass SCg we have σ = p σp hence σ = p σp . Finally each σp is a product of cycles, each of order |p|i appearing fσ (p)(i) times (Definition 15.7). Thus the partition of cycles of σp is |p| fσ (p) hence:     (79) σ = |p| fσ (p) = fσ (p) (−1)(|p|−1)g(p) = (−1) p (|p|−1)g(p) fσ (p) . p

For g ∈ Ξc set |g| := basic formula



p p (|p|

p

− 1)g(p). We are now ready to show the following the

15. MATRIX FUNCTIONS AND INVARIANTS

105

Theorem 15.20. Let g ∈ Ξc and let SCg be the corresponding superclass. Then  1  1  σ σ)) = σ Tfσ = (−1)|g| σg(p) (p). (80) Qf (T ( |Sf | |Sf | σ∈SCg

Proof. We have  σ Tfσ = σ∈SCg

σ∈SCg





σ Tfσ =

μ|s(μ)=g σ|fσ =μ

p∈W0







μ|s(μ)=g σ|σ∈Σμ



tμ(p) (p)

p∈W0

where the second equality follow from (77). By Proposition 15.11, Σμ is a Sf orbit under conjugation whose cardinality, by Proposition 15.12, is |Sf |/ p∈W0 zμ(p) . Moreover all elements σ ∈ Σμ have the same sign σ = μ := (−1)|g| p∈W0 μ(p) . Using (79) we then get,   tμ(p) (p)   tμ(p) (p) 1  σ Tfσ = μ = (−1)|g| μ(p) . |Sf | zμ(p) zμ(p) μ | s(μ)=g

σ∈SCg



= (−1)|g|

(81)

μ | s(μ)=g p∈W0

p∈W0

(



p∈W0 μ(p)g(p)

μ(p)

1 zμ(p)

tμ(p) (p)).

Finally by Formulas (78) and (21) we deduce    1 (−1)|g| ( μ(p) tμ(p) (p)) = (−1)|g| σg(p) (p), zμ(p) p∈W0 μ(p)g(p)

proving our claim.

p∈W0



Example 15.21. Take T = {1, 2, 3}, f : 1 → 1, 2 → 1, 3 → 2. The group Sf = {1, (1, 2)} has two double cosets in S3 , that is D1 := Sf , D2 := S3 \ Sf . We see that S3 \ Sf decomposes into 2 superclasses. One computes the functions 1  1 σ Tfσ = (σ1 (x1 )2 σ1 (x2 ) − σ1 (x21 )σ1 (x2 )) = σ2 (x1 )σ1 (x2 ), 2 2 σ∈D1

1  1 σ Tfσ = (2σ1 (x21 x2 ) − 2σ1 (x1 x2 )σ1 (x1 )) = σ1 (x21 x2 ) − σ1 (x1 x2 )σ1 (x1 ). 2 2 σ∈D2

Notice that by Formula (32), the sum of these two invariants is the degree 2, 1 part of σ3 (x1 + x2 ). 15.2. Proof of Theorem 1.10. Notice that Theorem follows TheΓ 1.10 Γ from orem 12.2 once we show that, for Γ = (γ1 , . . . γn ), θ ∈ ( V∗ ⊗ V )GL(V ) , we have tr(θ ◦ ∧γ1 ξ1 ⊗ . . . ⊗ ∧γn ξn ) is the evaluation of an element of S. ). Putting all the γi ’s together, we For each i = 1, . . . , n let γi = (γ1,i . . . , γri ,i n get a composition γ = (γ1,1 , . . . , γhn ,n )  N = i=1 hi . Let us see that for generic matrices ξj,i , i = 1, . . . n, j = 1, . . . , ri , we have that tr(θ ◦ ∧γ1,1 ξ1,1 ⊗ . . . ⊗ ∧γhn ,n ξhn ,n ) comes from S. Proposition 14.8, tells us that each such θ is a linear combination of the operators θD with D a double coset. Then decomposing each θD as a sum over the superclasses and applying Theorem 15.20, this follows from Formulas (71) and (77). We then deduce that, if we specialize ξj,i to ξi for each i = 1, . . . n, and j = 1, . . . , ri , then also tr(θ ◦ ∧γ1 ξ1 ⊗ . . . ⊗ ∧γn ξn ) comes from S. This is our claim. 

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IV. MATRIX FUNCTIONS AND INVARIANTS

15.2.1. The Theorem over Z. We claim now that the polynomials Z[xki,j ] on n-tuples of m × m matrices invariant under conjugation by GL(m, Z) are also generated over Z by the elements σi (M ), M ∈ W0 . This follows by the following argument. To start let us see that Q[xki,j ]GL(Q,m) = Q[xki,j ]SL(Q,m) = Q[xki,j ]SL(Z,m) . The first equality follows from the fact that the scalar matrices act trivially under conjugation. The second from the fact that SL(Z, m) is Zariski dense in the group SL(C, m). This is a standard fact, and a special case of a Theorem of Borel [7], let us sketch the proof for convenience. The main fact that we leave as exercise is that SL(C, m) is generated as group by the root subgroups ui,j (λ) := 1 + λei,j , i = j. This implies that there is a sequence of products ui1 ,j1 (λ1 ) . . . uiK ,jK (λK ) Zariski dense in the group SL(C, m). Then if the λi ∈ Z this is dense in this set of products and the claim follows. Next we observe that Z[xki,j ]SL(Z,m) = Q[xki,j ]SL(Q,m) ∩ Z[xki,j ] is a subring of Z[xki,j ] containing Z[σi (M )], M ∈ W0 and coinciding with it once we pass over Q. So if it does not coincide with Z[xki,j ]SL(Z,m) , necessarily there is a prime p such that reducing modulo p, F [σi (M )]  F [xki,j ]SL(F,m) , F any field of characteristic p. But, by Corollary 10.4 in each degree we have the equality of dimensions, dimF F [xki,j ]SL(F,m) = dimQ Q[xki,j ]SL(Q,m) = dimQ Q[σi (M )] = dimF F [σi (M )], M ∈ W0 . The claim follows.

Part V

Relations

108

V. RELATIONS

This part is entirely devoted to the proof of Theorem 1.12. 16. Relations We shall use the notations of Section 15. As usual we take a free module V, over a domain F , which is equal to Z or an infinite field, of rank m, for which we have chosen a basis so that we can identify it with F m . For an auxiliary N which will be taken large, we set W = V ⊕ F N −m . Definition 16.1. Let X = {x1 , , . . . , xn , . . .} be a finite or countable alphabet. We denote (by abuse of notations) by Sm X = Sm [ξ1 , . . . , ξn , . . .] the ring of d polynomial functions on the space Mm (F )X as in Formula (2) and by Sm X its degree d part. We denote by Fm X := Sm XGL(V ) the subring of Sm X consisting of those polynomial functions which are invariant under (simultaneous) conjugation, and by Fm Xd its degree d part. In 4.2.2 we have introduced the polynomial ring SF X in the variables σj (w), j > 0, w ∈ W0 a Lyndon word in the alphabet X (with a chosen total order). Notice that if X ⊂ X . we have obvious inclusions Sm X ⊂ Sm X , Fm X ⊂ Fm X , and SF X ⊂ SF X . The homomorphisms Em X : SF X → Fm X given by evaluation on m × m matrices (Formula 34) are compatible with the various inclusions and Theorem 1.10 and §15.2.1 assert that they are surjective. Definition 16.2. We denote the Kernel of Em X by Km X. ˜ m X equal to the ideal of SF X generated by all elements σk (A), k > We let K m, A ∈ F+ X and their polarized forms. Let us start by remarking that Theorem 1.12 is equivalent to the statement that ˜ m X and Km X coincide. Notice that, if we let Xn = {x1 , . . . xn } and the ideals K ˜ m Xn . ˜ m X = ∪n m. The second part is the object of section 18. We use the formulas (70) that describe linear generators for Aμγ and we prove, by a rather complicated sequence of ˜ m. combinatorial facts, that these generators give relations in the proposed kernel K

110

V. RELATIONS

17. Describing Km 17.1. A first description of the kernel. As we have seen already both SF and Fm are multigraded and the homomorphism Em preserves the multigrading. d d In particular it is degree preserving, so that it induces a map Em : SdF → Fm for d d d each degree d ≥ 0. Consequently Km = ⊕d Km with Km = ker Em . We start from a simple d = 0. Proposition 17.1. If d ≤ m, Km d d Proof. For any d we know by Theorem 1.10, that Em : SdF → Fm is surjective. d Also the dimension of SF is independent of F and so is, by Corollary 11.8, the d . Hence it is enough to prove our statement in characteristic 0 dimension of Fm where it is a consequence of Theorem 4.13. 

Assume now that we take d > m. Proposition 17.1 implies that, given any d (in particular with Fdd ). Moreover integer N ≥ d, we can identify SdF with FN d d since m < d, the surjective map SF → Fm is identified with the restriction map d d → Fm of invariants, of degree d, of n-tuples of N × N matrices (i.e. elements FN of End(W )) to invariants of degree d of n-tuples of m × m matrices, induced by the inclusion Mm (F ) ⊂ MN (F ). We are going to denote this map by E considering that we have fixed m once and for all. Let us now fix a multidegree c = (c1 , . . . , cn ) with |c| = d and let, for any integer c h > 0, denote by Sh X the multidegree c part of Sh X. Recall that in Section 12 we chose a total order on the set of sequences Λ = (λ1 , . . . λi ), λi a Young diagram c with ci boxes, and we defined a filtration of Sm X given by GL(V )2n -submodules MΛ (V ) (Formula (53)) such that, if for a given Λ, Λ denotes its predecessor, we have the exact sequence qΛ,V

0 → MΛ (V ) → MΛ (V ) −→ ∇Λ V ∗ ⊗ ∇Λ V → 0

(82) where

∇Λ V ∗ ⊗ ∇Λ V  ∇λ1 (V ∗ ) ⊗ ∇λ1 (V ) ⊗ ∇λ2 (V ∗ ) ⊗ ∇λ2 (V ) ⊗ · · · ⊗ ∇λn (V ∗ ) ⊗ ∇λn (V ) is a ∇-module for the group GL(V )2n . Setting Fm,Λ = MΛ (V )GL(V ) = MΛ (V ) ∩ Fm , we get a filtration of the multic degree c component Fm of the ring of invariants Fm . Moreover the sequence (82) is formed by modules with ∇-filtration with respect to the diagonal action of GL(V ) so that we have an induced exact sequence of invariants: GL(V ) qΛ,V

→ 0. 0 → Fm,Λ → Fm,Λ −→ ∇Λ V ∗ ⊗ ∇Λ V ˇ i , Γ = (γ1 , . . . , γn ) = Λ, ˜ we may consider Γ as a unique composition Set γi := λ of d, by putting together all the strings γi , and then Γ 

γn  V∗⊗ V.  Im(πZ,V ) Recall now that in Section 12 MΛ (V ) has been defined as MΛ (V ) = Z≤Λ ˜ with (83) ζ1 ζn Z Z Z      πZ,V : V ∗ ⊗ V = End( V ) → MΛ (V ), πZ,V (θ) = tr(θ ◦ ξ1 ⊗ . . . ⊗ ξn ).

V∗⊗

Γ 

V ∼ =

γ1 

V∗⊗

γ1 

V ⊗

γ2 

V∗⊗

γ2 

V ⊗ ...⊗

γn 

17. DESCRIBING Km

111

 Furthermore, denote by π ¯Z,V the restriction, to the invariants EndGL(V ) ( Z V ), of  πZ,V . By the proof of Theorem 12.2 we have Fm,Λ = Z≤Λ Im(¯ πZ,V ). ˜ Recall the definition of truncation Tr. Consider the torus TN −m ⊂ GL(V ) × GL(N − m) ⊂ GL(W ) consisting of pairs (id, t), t diagonal. Then given a GL(W )2n rational module M , denote by Tr(M ) the invariant space with respect to the torus TN2n−m . Thus, using Lemma 11.16 and Proposition 10.10 we deduce that Tr(

Γ 

W∗ ⊗

Γ 

W) =

Γ 

V∗⊗

Γ 

Tr(∇Λ W ∗ ⊗ ∇Λ W ) = ∇Λ V ∗ ⊗ ∇Λ V

V,

(including the trivial case in which Tr(



W∗⊗



W ) = Tr(∇Λ W ∗ ⊗ ∇Λ W ) = 0).

Lemma 17.2. Let N ≥ m. The map Tr : SN X → Sm X, coincides with the evaluation E when restricted to FN . In particular we have the commutative diagram

(84)

EndGL(W ) ( ⏐ ⏐ Tr



EndGL(V ) (



π ¯ Γ,W

qΛ,W

π ¯ Γ,V

qΛ,W

W ) −−−−→ FN,Λ −−−−→ ∇Λ W ∗ ⊗ ∇Λ W ⏐ ⏐ ⏐ ⏐ EΛ =Tr Tr V ) −−−−→ Fm,Λ −−−−→ ∇Λ V ∗ ⊗ ∇Λ V

Proof. By definition E consists in restricting the evaluation of the invariants to m × m matrices. Using the splitting W = V ⊕ F N −m , write an element B of End(W ) as a 2 × 2 block matrix

 B1,1 B1,2 B= , B2,1 B2,2 B1,1 ∈ End(V ). Thus applying E is equivalent to set equal to 0 the variables of the blocks Bi,j with (i, j) = (1, 1). This means projecting on the space Sm X which is exactly the 0 weight space for the torus TN2n−m .  Recall the exact sequences of modules with a ∇-filtration. (85) −1 0 −−−−−→ πΓ,W (MΛ (W )) −−−−−→ ⏐ ⏐ Tr −1 0 −−−−−→ πΓ,V (MΛ (V )) −−−−−→



W∗ ⊗ ⏐ ⏐ Tr



V∗⊗





qΛ,W ◦πΓ,W

W −−−−−−−−−→ ∇Λ W ∗ ⊗ ∇Λ W −−−−−→ 0 ⏐ ⏐ Tr V

qΛ,V ◦πΓ,V

−−−−−−−−→

∇Λ V ∗ ⊗ ∇Λ V

−−−−−→ 0

We claim that the second exact sequence is obtained from the first by truncation which is a surjective map. To see this, we only need to remark that by the exactness −1 (MΛ (W ))) equals the kernel of the map of Tr the space Tr(πΓ,W qΛ,V ◦ πΓ,V :

Γ 

V∗⊗

−1 and this coincides with πΓ,V (MΛ (V )).

Γ 

V → ∇Λ V ∗ ⊗ ∇Λ V

W∗ ⊗

0 −−−−→ S1 (V ) ⊗ S2 (V ) −−−−→







Figure 4

V)⊕(

W) ⊕ ( ⏐ ⏐ Tr

Figure 3





(S1 (V ) ⊗

0 −−−−→ S1 (W ) ⊗ S2 (W ) −−−−→ (S1 (W ) ⊗ ⏐ ⏐ Tr

−1 (MΛ (V ))GL(V ) 0 −−−−→ πΓ,V



qΛ,W ◦πΓ,W

V ∗ ⊗ S2 (V ))

−1 −−−−→ πΓ,V (MΛ (V )) −−−−→ 0

−1 W ∗ ⊗ S2 (W )) −−−−→ πΓ,W (MΛ (W )) −−−−→ 0 ⏐ ⏐ Tr

W )GL(W ) −−−−−−−→ (∇Λ W ∗ ⊗ ∇Λ W )GL(W ) −−−−→ 0 ⏐ ⏐ ⏐ ⏐ Tr Tr Γ ∗ Γ GL(V ) qΛ,V ◦πΓ,V −−−−→ ( V ⊗ V) −−−−−−−→ (∇Λ V ∗ ⊗ ∇Λ V )GL(V ) −−−−→ 0

−1 0 −−−−→ πΓ,W (MΛ (W ))GL(W ) −−−−→ ( ⏐ ⏐ Tr

112 V. RELATIONS

17. DESCRIBING Km

113

Let us now pass to invariants. Proposition 17.3. The commutative diagram shown in Figure 3 (see p. 112) has exact rows and surjective vertical arrows. Proof. By induction we may restrict to the case m = N − 1. We may assume λi , cf. Remark 11.17 (Λ = (λ1 , . . . λN )) equals to 0 that the N th component of N V = 0, the second line of the diagram is identically 0 and the otherwise, since statement is trivial. We then can apply Proposition 11.14 and deduce that the two maps Tr : (

Γ 



W ⊗

Γ 

W)

GL(W )

→(

Γ 



V ⊗

Γ 

V )GL(V ) ,

Tr : (∇Λ W ∗ ⊗ ∇Λ W )GL(W ) → (∇Λ V ∗ ⊗ ∇Λ V )GL(V ) are surjective. −1 −1 As for Tr : πΓ,W (MΛ (W ))GL(W ) → πΓ,V (MΛ (V ))GL(V ) , we take the spaces

S1 (W ) = Ker(

Γ 

S1 (V ) = Ker(

W ∗ → ∇Λ W ∗ ); S2 (W ) = Ker(

Γ 

V ∗ → ∇Λ V ∗ ); S2 (V ) = Ker(

Γ 

Γ 

W → ∇Λ W )

V → ∇Λ V ).

Thus we have a commutative diagram with exact rows as shown in Figure 3 (see p. 112) of modules with ∇-filtration as GL(W )2n modules, and as GL(V )2n modules respectively. Applying Proposition 11.14 we then deduce that the truncations Tr : (S1 (W ) ⊗ S2 (W ))GL(W ) → (S1 (V ) ⊗ S2 (V ))GL(V ) ; and Tr :((S1 (W ) ⊗

Γ 

W) ⊕ (

→ ((S1 (V ) ⊗

Γ 

Γ 

W ∗ ⊗ S2 (W )))GL(W )

V)⊕(

Γ 

V ∗ ⊗ S2 (V )))GL(V )

are surjective. Hence, by the snake Lemma, also −1 −1 (MΛ (W ))GL(W ) → πΓ,V (MΛ (V ))GL(V ) Tr : πΓ,W

is surjective, since taking invariants is an exact functor on modules having a ∇filtration.  Introducing notations for the different kernels and truncations we deduce (from the snake lemma) that the diagram in Figure 5 (see p. 114) has exact rows and columns.

KΛ,0 ⏐ ⏐ 

0

−1 0 −−−−→ πΓ,V (MΛ (V ))GL(V ) ⏐ ⏐ 

−1 (MΛ (W ))GL(W ) 0 −−−−→ πΓ,W ⏐ ⏐ Tr0 

0 −−−−→

0 ⏐ ⏐  qΛ,W ◦πΓ,W

0 ⏐ ⏐ 

Figure 5

0

0

KΛ,1 −−−−−−−→ KΛ,2 −−−−→ 0 ⏐ ⏐ ⏐ ⏐   Γ ∗ Γ q ◦π Λ,W Γ,W −−−−→ ( W ⊗ W )GL(W ) −−−−−−−→ (∇Λ W ∗ ⊗ ∇Λ W )GL(W ) −−−−→ 0 ⏐ ⏐ ⏐ ⏐ Tr1  Tr2    qΛ,V ◦πΓ,V −−−−→ ( Γ V ∗ ⊗ Γ V )GL(V ) −−−−−−−→ (∇Λ V ∗ ⊗ ∇Λ V )GL(V ) −−−−→ 0 ⏐ ⏐ ⏐ ⏐  

−−−−→

0 ⏐ ⏐ 

114 V. RELATIONS

17. DESCRIBING Km

115

We arrive at the final result of this section c

Theorem 17.4. The kernel of E = Tr restricted to FN is the sum of the spaces HΛ := πΓ,W (Ker (Tr1 )) = πΓ,W (KΛ,1 ). Proof. Since Tr ◦ πΓ,W = πΓ,V ◦ Tr (cf. Formula (84)) we have that HΛ ⊂ c Ker(E) ∩ FN . To show the reverse inclusion we proceed by induction on Λ. c Given a ∈ FN with Tr(a) = 0, take the minimum Λ such that a ∈ FN,Λ . We have Tr ◦ qΛ,W (a) = qΛ,V ◦ Tr(a) = 0 so that, with the notations of Figure 5 we have qΛ,W (a) ∈ KΛ,2 . By the exactness of the diagram in Figure 5 there is a c ∈ KΛ,1 such that qΛ,W ◦πΓ,W (c) = qΛ,W (a) that is a−πΓ,W (c) ∈ FN,Λ . By the definition of HΛ we have that πΓ,W (c) ∈ HΛ , moreover a − πΓ,W (c) ∈ FN,Λ ∩ Ker(Tr), therefore we finish by induction.  Remark 17.5. Observe that in Theorem 17.4 a special case is when the third row of the diagram in Figure 5 is identically 0. This happens when one of the partitions γi forming Γ has its first row > m. By the definition of the order on the partitions Λ this happens at the first layers of the filtration. In particular for the   very first layer all the γi = N and Γ W ∗ ⊗ Γ W is an irreducible module so that πΓ,W is an isomorphism to its image ∇Λ W ∗ ⊗ ∇Λ W . 17.2. More about truncation. Consider two partitions γ, θ  d. Take a vector space U over F and the module γ U . Finally let M be a GL(U ) module with a ∇-filtration and assume that M has a highest weight ≤ θ, by this we mean that we have a ∇-filtration {0 = H0 ⊂ · · · ⊂ Ht = M } so that each Mi /Mi−1 = ∇ζi (U ) and ζi ≤ θ (Definition 8.4 and Theorem 9.25). γ Proposition 17.6. Let φ ∈ homGL(U) ( U, M ) then φ is a linear combination μi ˇ of homomorphisms each factoring through some U with μi ≥ θ. Proof. We are going to make induction on t. If t = 1 then M = ∇ζ (U ) and we have an exact sequence ˇ

0 −→ T −→

ζ 

p

U −→ ∇ζ (U ) −→ 0, ζ ≤ θ. ˇ That is M = ∇ζ (U ) is a quotient of ζ U with kernel T with a ∇-filtration by Theorem 9.10. By Lemma 11.11 the following sequence is exact: ˇ

ζ γ γ    U, T ) → homGL(U) ( U, U ) → homGL(U) ( U, ∇ζ (U )) → 0. γ γ ζˇ Thus, given φ ∈ homGL(U) ( U, ∇ζ (U )), there exists φ ∈ homGL(U) ( U, U) ˇ such that φ = p ◦ φ , since ζ ≤ θ we have μ = ζˇ ≥ θ. Now if t > 1, we may assume that Im φ is not contained in Mt−1 by induction. If q : M → M/Mt−1  ∇ζ is the quotient, ζ ≤ θ, we have q ◦ φ = 0 and from the  ˇ previous case there is a φ ∈ homGL(U) ( γ U, ζ U ) such that q ◦ φ = p ◦ φ . By Lemma 11.11 again we have an exact sequence

0 → homGL(U) (

γ 

ˇ

0 → homGL(U) (

ζ 

ˇ

ζ 

ˇ

q◦

ζ 

U, Mt−1 ) → homGL(U) ( U, M ) −→ homGL(U) ( U, ∇ζ ) → 0. ζˇ It follows that there is b ∈ homGL(U) ( U, M ) such that q ◦ b = p hence q ◦ φ = p ◦ φ = q ◦ b ◦ φ

or

q ◦ (φ − b ◦ φ ) = 0.

116

V. RELATIONS

Thus Im(φ − b ◦ φ ) ⊂ Mt−1 and the claim follows by induction.



As before, dim V = m take the space W = V ⊕ F d−m . We have introduced in d−m from GL(W ) modules to GL(V ) Section 10.2 the truncation functor Tr := Tr0 modules. For the modules ∇μˇ one has from Proposition 10.10 ∇μˇ (V ) if μ1 ≤ m Tr(∇μˇ (W )) = . 0 if μ1 > m Take now, as in Proposition 11.14 Tr = Tr0 ⊗ Tr0 :

γ 

W ∗ ⊗ ∇μˇ (W )

γ γ γ    V ∗ ⊗ ∇μˇ (V ). = hom( W, ∇μˇ (W )) → hom( V, ∇μˇ (V )) = γ γ V and ∇μˇ (V ) are direct summands W and ∇μˇ (W ) respecNow recall that γ in W, ∇μˇ (W )), Tr(φ) = π ◦ φ ◦ i tively and we easily see that given φ ∈ homGL(W ) (   where i : γ V → γ W is the inclusion and π = Tr0 : ∇μˇ (W ) → ∇μˇ (V ) the projection. We have

Lemma 17.7. Let μ be a partition with μ1 ≤ m. Then, for every partition γ γ, the map Tr induces an isomorphism between homGL(W ) ( W, ∇μˇ (W )) and γ V, ∇μˇ (V )). homGL(V ) ( γ γ Proof. Since hom( W, ∇μˇ (W )) and hom( V, ∇μˇ (V )) have ∇-filtrations (as GL(W )and GL(V )-modules respectively), we deduce that the dimensions of  homGL(W ) ( γ W, ∇μˇ (W )) and of homGL(V ) ( γ V, ∇μˇ (V )) do not depend on the characteristic. To compute those dimensions let us work γ in characteristic 0. We get which that in both cases they equal the multiplicity of ∇μˇ in γ is the multiplicity of indeed independent the symmetric function sμˇ (x) in the character i σγi (x) of of V and W by our assumption on μ ˇ and Proposition 3.3. Take a highest weight vector v ∈ ∇μˇ (W ). Then v ∈ ∇μˇ (V ) and v spans the socle of ∇μˇ (W ) (resp. the socle γ of ∇μˇ (V )) as a GL(W ) (resp. GL(V )) module. W, ∇μˇ (W )) is non zero, there is a w such that Thus if φ ∈ homGL(W ) ( φ(w)= v. We can assume that the weight of w equals that of v. It follows that w ∈ γ V, and Tr(φ)(w) = v. Thus Tr is injective and by dimension considerations an isomorphism.  We can now show

 Proposition 17.8. Let φ ∈ EndGL(W ) ( γ W ), then Tr(φ) = 0 if and only if  φ is a linear combination of homomorphisms which factor through some μ W with μ1 > m. Proof. If Tr has such a factorization then Tr(φ) is clearly γ0. W . By Corollary For the converse, take a ∇-filtration 0 = M0 ⊂ · · · ⊂ Mt = 6.5 we may assume that there is an h such that each of the sub-quotients in the filtration 0 = M0 ⊂ · · · ⊂ Mh is one of the ∇μˇ with μ1 > m, while the induced filtration on Mt /Mh has sub-quotients ∇μˇ with μ1 ≤ m. Clearly Tr(Mh ) = 0 while, from lemma 17.7, the truncationTr induces γ γ an isoγ γ W, W/Mh ) and homGL(V ) ( V, V ). Thus morphism between homGL(W ) ( the truncation of a homomorphism is zero if and only if this homomorphism has

17. DESCRIBING Km

117

image in Mh . By Lemma 17.6, this implies that such a homomorphism is a linear  combination of homomorphisms which factor through some μ W with μ1 > m.   We now use the identification Φ : Aγ → EndGL(W ) γ W , cf. formula (69). We have, by Corollary 14.6 and Proposition 14.10, that Ker Tr is identified to the subspace Aμγ ⊂ Aγ (Definition 14.9). Furthermore the space Aμγ is formed by integral linear combinations of the elements described by Formula (70). 17.3. Conclusion. Let Λ be a sequence of partitions, and Γ the associated sequence of dual partitions, while γ is the decomposition formed by all the rows of all the partitions of Λ. d Theorem 17.9. The space Km of relations of invariants of m × m matrices in degree d is spanned by the elements πΓ,W ◦ Φ(Aμγ ), with W = F d and μ1 > m.

Proof. This is a consequence of Theorem 17.4 and Proposition 17.8. In fact d we may take N = d, W = F d and deduce that Km is the sum of the spaces Γ Γ W → EndGL(V ) V. HΛ = πΓ,W Ker Tr, with Tr : EndGL(W ) γ Γ W = W and we finally have that, under the identification By definition   Φ : Aγ → EndGL(W ) γ W , the space Ker Tr is identified to Aμγ .

118

V. RELATIONS

˜m 18. Km versus K 18.1. The elements c(a, k). Let p be a refinement of k that is k = q ◦ p hence Sp ⊂ Sk . Let a ∈ Q[SN ] be an element invariant under conjugation with respect to Sp .  Lemma 18.1. 1) The element x∈Sk /Sp xax−1 is well defined independently of the  choice of the coset representatives x ∈ Sk /Sp . 2) The element x∈Sk /Sp xax−1 is invariant under conjugation with respect to Sk . 3) We have,  1 1 T (a))). Qk (T ( xax−1 )) = Qq (Qp ( |Sk | |Sp | x∈Sk /Sp

 the invariance. As for 2), if g ∈ Sk we have that  Proof. 1) is clear from g x∈Sk /Sp xax−1 g −1 = gx∈Sk /Sp gxa(gx)−1 and the statement is a consequence of 1). Finally for the third statement we have that, since x ∈ Sk , Qk (T (xax−1 )) = Qk (T (a)) so  1 1 |Sk | 1 Qk (T (a)) = Qq (Qp (T (a))) Qk (T ( xax−1 )) = |Sk | |Sk | |Sp | |Sp | x∈Sk /Sp



and the formula follows.

Take the composition γ  N with the associated surjection k : [1, . . . N ] → [1, . . . , h] and the subgroup Sk of permutations in SN which commute with k.  For a = ασ σ ∈ Q[SN ] we introduce the notation: 1  (86) c(a, k) := ασ Tkσ . |Sk | σ∈SN

Notice that by (77), we have the equalities (87)

c(a, k) =

1 Qk (T (a)). |Sk |

Given g ∈ Ξγ , g defines a Young superclass SCg . By Theorem 15.20   c(g) := (−1)|g| σg(w) (w) = c( σ σ, k). w∈W0

σ∈SCg

Since by Proposition 15.19, any double coset Dσ,k = Sk σSk is a union of superclasses, any element a ∈ Z[SN ] with the property that τ aρ = τ ρ a for τ, ρ ∈ Sk , can be written as   rg σ σ, a= g∈Ξγ

σ∈SCg

rg ∈ Z. So, for such an a: Lemma 18.2. c(a, k) =

 g∈Ξd

In particular c(a, k) ∈ SZ .

rg c(g).

˜m 18. Km VERSUS K

119

Let us now look at the behaviour of c(a, k) under polarization. Assume that a is invariant under conjugation by Sk . In this case we have, by Proposition 13.6.2, that 1 Pk ◦ Qk (T (a)) = Pk (c(a, k)). T (a) = |Sk | Let p : [1, . . . N ] → [1, . . . , s] be a refinement of k and q : [1, . . . s] → [1, . . . , h] such that k = q ◦ p, p the corresponding sequence. From the very definitions we deduce p

Lemma 18.3. Let Pq be the map of partial polarization as in Definition 13.3. Then p

c(a, p) = Pq (c(a, k)).

(88)

Proof. We have by Formula (87) c(a, k) = |S1k | Qk (T (a)). By definition T (a) is a multilinear element of S and, by hypothesis a is invariant under conjugation by Sk . So from Formula (60): p

T (a) = Pk (c(a, k)) = Pp ◦ Pq (c(a, k)) so that

p

p

Qp (T (a)) = Qp ◦ Pp ◦ Pq (c(a, k)) = |Sp |Pq (c(a, k)). This gives c(a, p) =

1 p Qp (T (a)) = Pq (c(a, k)). |Sp |



18.2. A basic reduction. In view of Theorem 17.9, in order to prove Theorem 16.3, we need to show that the elements πΓ,W ◦ Φ(Aμγ ), W = F d with μ1 > m ˜ m of SF generated by all evaluations σk (A), k > m, A ∈ F+ X belong to the ideal K and their polarized forms, as in Definition 16.2. At this point remark that if γ = (γ1,1 , . . . γ1,h1 , . . . , γn,1 , . . . γn,hn ) with γi = . . . γi,hi ), πΓ,W ◦ Φ(Aμγ ) is the restriction of the function πγ,W ◦ Φ(Aμγ ) on (γi,1 ,  H = hi -ples of m × m matrices (A1 , . . . AH ) to those H-ples where Ah0 +···+hi−i +1 = Ah0 +···+hi−i +2 = · · · = Ah0 +···+hi−i +hi , h0 = 0, i = 1, . . . , n. We therefore need to study the elements πγ,W ◦ Φ(θ) with θ ∈ Aμγ , γ, μ, two compositions of d associated to two maps k, f and described by Formula (70) with μ1 > m. By Proposition 14.10 we may take    gD1 ,D2 , gD1 ,D2 = (89) θ= σ1 σ2 τ σ1 τ σ2−1 , |Sk | σ1 ∈L(D1 ) σ2 ∈L(D2 )

τ ∈Sf

where Di are double cosets with respect to the action of Sk × Sf , Sk acting by left multiplication and Sf by right multiplication, and with L(Di ) left transversals for  Sf , (i.e. Di = ˙ σSf ). σ∈L(Di )

Now remark that by Formulas (77) and (55) we deduce that if k is the map giving the composition γ: (90)

πγ,W ◦ Φ(θ) = Qk (T (θ)) = c(gD1 ,D2 , k)

Thus Theorem 16.3 is a consequence of

120

V. RELATIONS

Theorem 18.4. If μ1 > m the element c(gD1 ,D2 , k) lies in the ideal (cf. 16.2) ˜ Km ⊂ SZ . Theorem 18.4 will follow from the following special case. Take σ ∈ SN and define an equivalence relation on [1, . . . , N ] by setting u  v if f (u) = f (v) and f (σ −1 (u)) = f (σ −1 (v)). If we denote for each i = 1, . . . , h, Ci := f −1 (i), the equivalence classes are the non empty sets of the form Ci ∩ σ(Cj ). By indexing these equivalence classes with numbers 1, . . . , r, to this equivalence relation there corresponds a surjection b : [1, . . . , N ] → [1, . . . , r] for which we have Sb = Sf ∩ σSf σ −1 = Sf ∩ Sfσ , where to simplify notations we set, for a permutation π and a subgroup H in SN , H π := πHπ −1 . We have two obvious double cosets Sf and σSf with respect to Sb and Sf acting by left and right multiplication respectively. We can take L(Sf ) = {1}, L(σSf ) = {σ} and set:  (91) Φf,σ := gSf ,σSf = ( τ τ )σ −1 . τ ∈Sf

were the last equality is a consequence of (89). Theorem 18.4 then follows from Proposition 18.5. If μ1 > m then for any σ ∈ SN , the element c(Φf,σ , b) lies ˜ m. in the ideal K Remark 18.6. The element c(Φf,σ , b) is in SZ . Proof. We clearly have that ρΦf,σ γ = ρ γ Φf,σ , ∀ρ ∈ Sf , γ ∈ Sfσ , so Φf,σ and b satisfy the hypotheses of the element a and the map k of lemma 18.2 hence the claim.  We postpone the proof of Proposition 18.5 to §18.3 and show how this special case (where k = b) implies our Theorem 18.4. Proof. Proposition 18.5 implies Theorem 18.4. The diagonal action of Sk on SN /Sf × SN /Sf preserves the subset D1 /Sf × D2 /Sf , which is thus partitioned into Sk orbits O1 , . . . Op . We are now going to carefully choose the subset L(D1 ) × L(D2 ) which is in bijection with D1 /Sf × D2 /Sf . For each i = 1, . . . , p, choose a pair of cosets (σ1,i Sf , σ2,i Sf ) ∈ Oi . The orbit Oi is then in 1–1 correspondence with the set Sk /Ski of left cosets of Ski in Sk , where Ski is the stabilizer in Sk of the pair (σ1,i Sf , σ2,i Sf ) ∈ Oi . Hence: (92)

σ

σ

Ski = Sk ∩ Sf 1,i ∩ Sf 2,i .

We can take for ki a surjection associated to the equivalence relation on [1, . . . , N ] −1 −1 −1 −1 (j)) = f (σ1,i ()) and f (σ2,i (j)) = f (σ2,i ()). given by j   if k(j) = k(), f (σ1,i Now choose a set Ji ⊂ Sk of coset representatives of Sk /Ski . We can then take as L(D1 ) × L(D2 ) as the disjoint union ∪pi=1 {(xσ1,i , xσ2,i )|x ∈ Ji }. If we define, for each i = 1, . . . p :   −1 −1 xσ1,i ( τ τ )σ2,i x , gOi := σ1,i σ2,i x∈Ji

τ ∈Sf

˜m 18. Km VERSUS K

121

from Formula (89) we then have: gD1 ,D2 =

p 

gOi so that c(gD1 ,D2 , k) =

i=1

p 

c(gOi , k).

i=1

−1 From the definition of Formula (92) it follows that σ1,i Sf σ2,i is invariant under conjugation by Ski . Since ki is a refinement of k, we can apply Lemma 18.1, with −1 p = ki and a = σ1,i Sf σ2,i . We deduce that

(93)

c(gOi , k) = Qk (c(σ1,i (



−1 τ τ )σ2,i , ki )).

τ ∈Sf

 −1 It follows that, if for each i = 1, . . . , p, c(σ1,i ( τ ∈Sf τ τ )σ2,i , ki ) lies in the ideal ˜ m also c(gO , k) and hence c(gD ,D , k) lie in K ˜ m . So one needs to show that K i 1 2 Proposition 18.5 implies this. Let us simplify the notations, fix i and set σ1 = σ1,i , σ2 = σ2,i , define the surjection b whose group is Sb = Sfσ1 ∩ Sfσ2 ⊃ Ski . Thus ki is also a refinement of b and, letting k i be the corresponding sequence, we have by Lemma 18.3, to b in place of k and Formula (88) gives:   k c(σ1 ( τ τ )σ2−1 , ki ) = Pbi (c(σ1 ( τ τ )σ2−1 , b )). τ ∈Sf

τ ∈Sf k

˜ m is closed under the polarization operator P i we are reduced to show Since K b that Proposition 18.5 implies that  ˜ m. τ τ )σ2−1 , b ) ∈ K c(σ1 ( τ ∈Sf

Let σ = σ1−1 σ2 . Then Sb = Sfσ1 ∩ Sfσ2 = Sfσ1 ∩ Sfσ1 σ = (Sf ∩ Sfσ )σ1 and 

(94)

τ σ1 τ σ2−1 = σ1 (

τ ∈Sf



τ τ σ −1 )σ1−1 .

τ ∈Sf

If we now take the surjection b associated to the equivalence relation on [1, . . . , N ] given by j  , if f (j) = f () and f (σ −1 (j)) = f (σ −1 ()), we get that c(



τ τ σ −1 , b) = c(Φf,σ , b),

σ −1

Sb = Sf ∩ Sfσ = Sb1 .

τ ∈Sf

˜ m , also Using (94) we then deduce that if c(Φf,σ , b) ∈ K  ˜m τ σ1 τ σ2−1 , b ) ∈ K c( τ ∈Sf

proving that Proposition 18.5 indeed implies Theorem 18.4.



122

V. RELATIONS

18.3. Proof of Proposition 18.5. In this section we are going to prove Proposition 18.5. The proof will be achieved through a sequence of steps. We start by recalling and introducing some notation. The sequence μ = (μ1 , . . . μh ) is associated to the surjection f : [1, . . . , N ] → [1, . . . , h] by setting μi := |f −1 (i)|. We assume μ1 > m. Given j = 1, . . . h, a set Cj := f −1 (j) ⊂ [1, . . . , N ] will be called a layer. So Sf := {τ ∈ SN | f ◦ τ = f } =

h 

Sf −1 (j) =

i=1

h 

SCi

i=1

is the subgroup which preserves all the layers. Conjugating by σ, we see that σSf σ −1 = {τ ∈ SN | f ◦ σ −1 τ σ = f } = {τ ∈ SN | f ◦ σ −1 ◦ τ = f ◦ σ −1 } = Sf ◦σ−1 . We also consider a surjection b : [1, . . . , N ] → [1, . . . , r] indexing the nonempty subsets Cs ∩ σ(Ct ), s, t ∈ [1, . . . , h]. It follows that b(i) = b(j) if and only if f (i) = f (j) and f ◦ σ −1 (i) = f ◦ σ −1 (j). Furthermore, Sb = Sf ∩ Sfσ and    (95) Sb = SCa ∩σ(Cc ) = SCa ∩ Sσ(Cc ) . a,c∈[1,...,h] | Ca ∩σ(Cc )=∅

a∈[1,...,h] c∈[1,...,h]

18.3.1. Three reduction steps. Let us start by remarking that, since our element Φf,σ depends up to a sign, only on the coset σSf , we can change σ by right multiplication by an element in Sf . We can thus normalize σ using the following: Step 18.1 (Splitting cycles). For every element w ∈ SN there is a τ ∈ Sf such that, if we write σ := wτ = c1 · · · ct as a product of disjoint cycles, we have that for any j, if cj = (a1 , . . . aj ), the restriction f|{a1 ,...,aj } is injective. Proof. This is similar to Lemma 4.11. Assume there is a cycle cj = (a1 , . . . , aj ) with f (a1 ) = f (aq ) for q > 1. Then the transposition (a1 , aq ) ∈ Sf and cj (a1 , aq ) =  (a1 , aq+1 , . . . , aj )(a2 , . . . , aq ). An easy induction then implies our claim. Thus from now on we shall assume that we have chosen σ = c1 · · · ct such that for any j, if cj = (a1 , . . . aj ), then the restriction f|{a1 ,...,aj } is injective. Remark 18.7. For any N , if f is constant, we have Sf = SN and we can take σ = id and b = f . By Formula (80), which is in fact a reinterpretation of the basic Formula (21), we have that: c(Φf,σ , b) = σN (x1 ). Hence as soon as N ≥ m + 1, Proposition 18.5 holds for a constant f . In particular if N = m + 1 the only possible f , with μ1 > m is the constant map. This allows us to make induction on N ≥ m + 1 and also assume that f is non constant. Definition 18.8. The pair (f, σ) is called indecomposable is there is no proper, non empty subset H ⊂ [1, . . . , h] such that σ(f −1 (H)) = f −1 (H). Step 18.2. It suffices to show Proposition 18.5 for indecomposable pairs. Proof. Assume that (f, σ) is decomposable. This means that we can find ∅  H  [1, . . . , h], 1 ∈ H such that σ(f −1 (H)) = f −1 (H). Set A = f −1 (H) and B = [1, . . . , N ] \ A. Denote by f1 (resp. f2 ) the restriction of f to A (resp. B) so

˜m 18. Km VERSUS K

123

Sf = Sf1 × Sf2 . We can write σ = σ1 σ2 with σ1 ∈ SA , σ2 ∈ SB . We immediately get    Φf,σ = τ τ σ −1 = ( τ τ σ1−1 )( τ τ σ2−1 ) = Φf1 ,σ1 Φf2 ,σ2 . τ ∈Sf

τ ∈Sf1

τ ∈Sf2

Furthermore, if we let b1 (resp. b2 ) be the restriction of b to A (resp. B), we get Sb1 = Sf1 ∩ Sfσ11 , (resp. Sb2 = Sf2 ∩ Sfσ22 ), Sb = Sb1 × Sb2 implies |Sb | = |Sb1 | × |Sb2 | and we obtain c(Φf,σ , b) = c(Φf1 ,σ1 , b1 )c(Φf2 ,σ2 , b2 ). ˜ m which implies Now |f −1 (H)| < N and by induction we get that c(Φf1 ,σ1 , b1 ) ∈ K ˜  that c(Φf,σ , b) ∈ Km , since by Remark 18.6 we have that c(Φf ,σ , b2 ) ∈ SZ . 2

2

So from now on, we assume f, σ indecomposable. In particular, given any layer C, we have C = σ(C) so that C ∩ σ(C) = C. Our third step is crucial Step 18.3. It suffices to prove Proposition 18.5 assuming that, for any i, j satisfying f (i) = f (j) > 1, we have f (σ(i)) = f (σ(j)). Remark 18.9. Notice that this means that, given a layer Cs , s = 1, there is a layer Ct , t = s with σ(Cs ) ⊂ Ct or equivalently Cs ⊂ σ −1 (Ct ). So under this reduction, the set Cj ∩ σ(Ci ) for i > 1 is either empty or equal σ(Ci ). Thus the layers of the map b are:   (96) σ(Ci ), i = 1, σ(C1 ) ∩ Ci = ∅, Sb = Sσ(Ci ) × Sσ(C1 )∩Ci . i=1

i

Proof. We are going to make increasing induction on the number h − 1 of layers, Cs , s > 1. If all of them satisfy our assumption there is obviously nothing to prove. In particular, since f, σ indecomposable, this happens if |Cs | = 1 ∀s > 1, that is if h − 1 = N − |C1 |.  For every layer C, we have C = hj=1 (C ∩ σ −1 (Cj )) and by assumption 18.2 C ∩ σ(C) = C. Suppose that there is 1 < k ≤ h such that the layer C = Ck is the −1 disjoint union of the two non empty subsets A = C ∩σ (Cj ), j = k and B = C \A. ¯ ¯ Write Sf = SC × Sf , with Sf = j=k SCj . We define a new equivalence relation  on [1, . . . , N ] splitting the layer C in the two layers A, B by setting ⎧ ⎪ ⎨f (i) = f () = k or i   if i and  ∈ A or ⎪ ⎩ i and  ∈ B We denote by f : [1, . . . , N ] → [1, . . . , h + 1] a corresponding surjection. f is a refinement of f , so Sf  = SA × SB × S¯f ⊂ Sf = SC × S¯f and furthermore Sf /Sf  = SC /SA × SB . Next we consider the group Sb = Sf ∩ Sfσ . Sb also decomposes (by Formula (95)) as (Sb ∩ SC ) × (Sb ∩ S¯f ) and acts on Sf /Sf  with Sb ∩ S¯f acting trivially. We then choose coset representative for Sf /Sf  as follows. We first divide Sf /Sf  into its Sb , or equivalently Sb ∩ SC , orbits O1 , . . . , Op . In each orbit Oj , j = 1, . . . , p we choose a coset wj Sf  with wj ∈ SC . It follows w that Sf j = (SA × SB )wj × S f so that w

w

Oj = Sb /(Sb ∩ Sf j ) = (Sb ∩ SC )/(Sb ∩ SC ∩ Sf j ),

124

V. RELATIONS

so that the orbit Oj consists of the cosets v0,j wj Sf  , . . . vqj ,j wj Sf  where the vi,j ’s are suitable representatives in Sb ∩ SC for the cosets modulo the stabilizer Sb ∩ SC ∩ w Sf j of wj Sf  . Thus Φf,σ =

p 

wj ΦOj ,σ , ΦOj ,σ :=

j=1

qj 



v,j v,j wj

τ τ σ −1 .

τ ∈Sf 

=0

˜ m it suffices to see that, for each j = Thus, in order to prove that c(Φf,σ , b) ∈ K ˜ 1, . . . p, c(ΦOj ,σ , b) ∈ Km . We now claim that Lemma 18.10. If v ∈ Sb ∩ SC then σ −1 vσ ∈ Sf  . In particular Sb ∩ SC ∩ Sfw = Sfσ ∩ Sfw ∩ SC for any w ∈ SC . Proof. Indeed, since v ∈ Sb ⊂ σSf σ −1 we have σ −1 vσ ∈ Sf so, in order to prove that σ −1 vσ ∈ Sf  , it suffices to show that σ −1 vσ preserves A. Now if a ∈ A we have σ(a) ∈ Cj and Cj is disjoint from C hence, since v ∈ SC , vσ(a) = σ(a) and thus σ −1 vσ(a) = a. To finish, the first part gives Sb ∩SC ⊂ Sfσ , so that Sb ∩SC ∩Sfw ⊆ Sfσ ∩Sfw ∩SC . On the other hand, w ∈ SC , so that Sfσ ∩ Sfw ⊂ Sfσ ∩ Sf = Sb .  We deduce that, if v ∈ Sb ∩ SC :     v τ τ = ( τ τ )σ −1 v −1 σ so that v τ τ σ −1 = ( τ τ )σ −1 v −1 . τ ∈Sf 

τ ∈Sf 

τ ∈Sf 

τ ∈Sf 

Thus, for all j we have, since for all , v,j ∈ Sb ∩ SC : ΦOj ,σ =

qj  =0

v,j v,j wj



τ τ σ −1 =

τ ∈Sf 

qj  =0

v,j (



−1 wj τ τ σ −1 )v,j .

τ ∈Sf 

We claim that the subgroup Sb ∩ Sfw stabilizer of the coset wSf  in Sb equals Sfσ ∩Sfw . For this it is enough to show that these two Young subgroups fix the same decomposition of [1, . . . , N ]. Clearly Sb ∩ Sfw = Sfσ ∩ Sfw is the subgroup stabilizing the refinement of the two partitions σ(Ci ), i = 1, . . . , h and Ci , i = k, w(A), w(B) while Sfσ ∩ Sfw is the subgroup stabilizing the refinement of the two partitions σ(Ci ), i = k, σ(A), σ(B) and Ci , i = k, w(A), w(B) Notice that σ(C) ∩ Cj = σ(A), σ(B) ∩ Cj = ∅, while for i = j, σ(C) ∩ Ci = σ(B)∩Ci . It then follows that the two refinements coincide and the claim is proved. Thus   x( wj τ τ σ −1 )x−1 . (97) ΦOj ,σ = x∈Sb /(Sfw ∩Sfσ )

τ ∈Sf  w

Let b j denote the surjection whose corresponding group is Sf j ∩ Sfσ . Since ∩ Sfσ ⊂ Sb , b j is a refinement of b. As before we claim that we can use Lemma   18.1 for a = τ ∈Sf  wj τ τ σ −1 , k = b, p = b j , in fact a = τ ∈Sf  wj τ τ σ −1 is w Sf j

w

w

invariant under conjugation by Sf j ∩ Sfσ since τ aρ = τ ρ a, τ ∈ Sf j , ρ ∈ Sf  . By ˜ m it is enough to verify that: Formula (97), in order to show that c(ΦOj ,σ , b) ∈ K  ˜ m. c( τ wj τ σ −1 , b j ) ∈ K τ ∈Sf 

˜m 18. Km VERSUS K

125

We then set σj = wj−1 σ. We have Sf j ∩ Sfσ = Sf j ∩ Sf j Furthermore   τ wj τ σ −1 = wj ( τ τ σj−1 )wj−1 . w

τ ∈Sf 

w

w σj

σ

= (Sf  ∩ Sf j )wj .

τ ∈Sf  σ

So, if we let bj denote the surjection whose associated group is Sf  ∩ Sf j , we deduce  ˜ m if c(Φf  ,σ , bj ) ∈ K ˜ m. that c( τ ∈Sf  τ wj τ σ −1 , b j ) ∈ K j ˜ m ,, using the induction hyAt this point we can show that c(Φf  ,σj , bj ) ∈ K

pothesis. Indeed, the first layer of f coincides with the first layer of f and f has one more layer than f . So, either f , σj satisfy the conditions of our proposition 18.3 or we can continue. But this procedure must stop at some point.  18.3.2. End of the proof of Proposition 18.5. In view of what we has seen, we take a pair (f, σ) with f : [1, . . . , N ] → [1, . . . h] a surjection such that |C1 | > m and σ ∈ SN . We assume 1) For each cycle c = (i1 , . . . , is ) of σ, f|{i1 ,...is } is injective. 2) The pair (f, σ) is indecomposable. 3) For all k > 1, f|σ(Ck ) is constant. Remark that clearly assumptions 2) and 3) imply that the set of elements fixed by σ is contained in C1 . Furthermore using Remark 18.7, we assume h ≥ 2. Let us now define a function p : [2, . . . , h] → [1, . . . , h] by setting for each k > 1, p(k) := f (σ(Ck )), i.e. σ(Ck ) ⊂ Cp(k) , ∀k = 1. Remark first that, by assumption 2) we always have p(k) = k. If, for k ≥ 2, we have p(k) ≥ 2 then we can iterate this function and consider p2 (k). So when we write pi (k) we implicitly mean that all the elements pj (k) ≥ 2 for all j < i. Lemma 18.11. For each 2 ≤ k ≤ h there exists a minimum integer ht(k) ≥ 1 such that pht(k) (k) = 1. Furthermore pj (k) = p (k) for 0 ≤ j <  ≤ ht(k). We call ht(k) the height of k. Proof. This follows from the indecomposability of the pair (f, σ). Indeed by construction, for k > 1 we have σ(Ck ) ⊂ Cp(k) . So, if our statement is not true, there is an index a > 1 and some j > 1 with pi (a) ≥ 2, ∀i ≤ j, pj (a) = a. Since by definition σ : Ck → Cp(k) , ∀k ≥ 2 we necessarily have |Ck | ≤ |Cp(k) |, for each k ≥ 2. Thus, if pj (a) = a for 0 ≤ i < j we must have |Cpi (a) | = |Cpi+1 (a) | and hence σ(Cpi (a) ) = Cpi+1 (a) . We deduce that the set ∪j−1 i=0 Cpi (a) is σ stable and proper, contrary to the hypothesis.  Choose k ≥ 2 of maximal height r ≥ 1 and let s = p(k). Clearly if j ≥ 2 we must have Ck ∩ σ(Cj ) = ∅ otherwise, by assumption 3) we would have Ck ⊃ σ(Cj ) and thus k = p(j) would not have maximal height. Thus Ck ⊂ σ(C1 ). Therefore for all j > 1 with p(j) = s, we have Cj ⊂ σ(C1 ) and ht(j) = r. We now take A := {j ∈ [2, . . . , h] | p(j) = s}, so  Cj . f −1 (A) = j|σ(Cj )⊂Cs

We have just seen that one has σ

−1

(f

−1

(A)) ⊂ C1 . If we set

K := [1, . . . , N ] \ f −1 (A) = f −1 ([1, . . . , h] \ A) =

 j ∈A /

Cj ,

126

V. RELATIONS

K is a union of layers for f , including C1 and Cs (with the possible case 1 = s and r = 1 included). This decomposition of K is determined by the restriction f : K → [1, . . . , h] \ A, of f to K. Observe next that, since σ(f −1 (A)) ⊂ Cs we have σ(f −1 (A)) ∩ f −1 (A) = ∅, therefore we can define a permutation w ∈ SN by  w = w−1 := (j, σ(j)). j∈f −1 (A)

We then set σ := wσ and notice that Lemma 18.12. 1) σ = σ on the complement of f −1 (A) ∪ σ −1 (f −1 (A)).

2) σ is the identity on f −1 (A). In particular σ preserves K. 3) As a permutation on K, σ coincides with σ 2 on the set σ −1 (f −1 (A)) ⊂ C1 and with σ on its complement. 4) Cs ⊂ σ (C1 ). Proof. 1) By definition w is the identity on the complement of σ(f −1 (A)) ∪ f (A), therefore σ = σ on the complement of f −1 (A) ∪ σ −1 (f −1 (A)). 2) On f −1 (A) σ coincides with w, so since w2 = 1, we have that σ is the identity on f −1 (A). 3) Next σ maps σ −1 (f −1 (A)) to f −1 (A) where σ coincides with w, hence

σ = wσ coincides with σ 2 on σ −1 (f −1 (A)). 4) We have Cs = ∪hi=1 (Cs ∩ σ(Ci )) = σ(f −1 (A)) ∪ (Cs ∩ σ(C1 )) since Cs ∩ σ(Ci ) = ∅ if i = 1 or i ∈ / f −1 (A). Take r ∈ Cs . If r ∈ σ(f −1 (A)), take σ −2 r ∈ σ −1 (f −1 (A)) ⊂ C1 . By 3), r = σ 2 (σ −2 r) = σ (σ −2 r) ∈ σ (C1 ). If r ∈ Cs ∩ σ(C1 ), then by 1) w is the identity on r so that r = wr ∈ wσ(C1 ) = σ (C1 ).  −1

Lemma 18.13. c(Φf,σ , b) = |Γ|c(Φf  ,σ w, b),

(98)

Proof. Set Sf  = remark that

j ∈A /

SCj and Γ :=

σΓσ −1 =



Γ :=



SCj .

j∈A

j∈A

SCj so that Sf = Sf  × Γ. Now

Sσ(Cj ) ⊂ SCs ⊂ Sf  .

j∈A

Thus it is clear that σΓσ −1 ⊂ Sf  ∩ Sfσ ⊂ Sb = Sf ∩ Sfσ . Using Definition 15.9 and applying Lemma 15.10 for b in place of f we have:      Φf,σ = τ τ σ −1 = γ ( τ τ )γσ −1 = ρ ( τ τ )σ −1 ρσσ −1 τ ∈Sf

=



ρ∈σΓσ −1

γ∈Γ

ρ

−1

(



τ ∈Sf 

τ ∈Sf 

τ τ )σ

−1

ρ ≡b |Γ|



ρ∈σΓσ −1

τ τ σ

τ ∈Sf 

w = |Γ|Φf  ,σ w.

τ ∈Sf 

Therefore by Lemma 15.10, Formula (98) follows.



−1





We have seen that σ permutes the elements of K and also Sf  = j ∈A / SCj does. Let p denote the restriction of b to K. Since for each j ∈ A we have that Cj = Cj ∩ σ(C1 ), each Cj , j ∈ A, is a layer of the function b so by definition or by Formula (96) Sb = Sp × Γ. Let us now consider a function b on K whose

˜m 18. Km VERSUS K

127



corresponding group is Sf  ∩ Sfσ . Observe that |K| < N and f −1 (1) = f −1 (1) has ˜ m. more than m + 1 elements, so that by induction we have that c(Φf  ,σ , b ) ∈ K ˜ m. Lemma 18.14. p is a refinement of b and c(Φf  ,σ , p) ∈ K Proof. It suffices to show that p is a refinement of b and use Lemma 18.3 ˜ m is closed under polarization. (for b in place of k) and finally use the fact that K /A The partition associated to p is formed by decomposing the layers Ct , t ∈ by taking Ct ∩ σ(Cj ), ∀j. The one associated to b is formed by decomposing the / A by taking Ct ∩ σ (Cj ), ∀j ∈ / A. layers Ct , t ∈ If j ∈ / A and j = 1 we have, by Lemma 18.12 1), σ (Cj ) = σ(Cj ) so the layer of b : Ct ∩ σ (Cj ) = Ct ∩ σ(Cj ) is also a layer of p. There remains a unique layer  Ct ∩ σ (C1 ) for b which is thus the union of the remaining layers of p. Consider the map rw : Zxi i∈K → Zxj j=1,...,N defined by xj xσ−1 (j) if j ∈ σ(f −1 (A)) ⊂ Cs rw (xj ) = xj otherwise which induces, by substitution, a map tw : SZ xi i∈K → SZ xj j=1,...,N . ˜ m . We claim first that ˜ m also tw (F ) ∈ K Remark that if F ∈ K Lemma 18.15. Two elements i, j ∈ σ(f −1 (A)) are equivalent with respect to b if and only if σ −1 (i) and σ −1 (j) are equivalent with respect to f . Proof. In fact σ(f −1 (A)) decomposes into the sets σ(C ),  ∈ A and each  σ(C ) = σ(C ) ∩ Cs is a layer for b. Lemma 18.15 then implies that the map ψ on the set b(σ(f −1 (A)) defined by choosing i ∈ σ(f −1 (A)) such that a = b(i) and then setting ψ(a) := b(σ −1 (i)) is well defined and depends only upon a and not on the choice of i. As a consequence, b on the variables xa , a ∈ b(K) defined by we get a map rw xa xψ(a) = xb(i) xb(σ−1 (i)) if a = b(i), i ∈ σ(f −1 (A)) b (99) rw (xa ) = otherwise. xa b induces then a map map tbw : Sxa a∈b(K) → Sxa a∈b([1,...,N ]) on the The map rw corresponding algebras. In particular we claim that, using Formula (77):

Lemma 18.16. If  ∈ SK we have Qb (Tw ) = tbw (Qp T ). Proof. The permutation w is obtained from  by substituting, in each cycle of the cycle decomposition of , each element i such that σ −1 (i) ∈ f −1 (A) with the string i, σ −1 (i). Thus we have, using the map σ → Tσ , that Tw = tw (T ). Thus Qb (Tw ) = Qb (tw (T )) = tbw (Qb T ) = tbw (Qp T ), since Qb (T ) = Qp (T ) because p = b on K.



128

V. RELATIONS

We can now finish the proof of Proposition 18.5. Indeed, from Formula (98) and Lemma 18.16 we get, c(Φf,σ , b) = |Γ|c(Φf  ,σ w, b) =

|Γ|  1  τ Qb Tτ σ −1 w = τ Qb Tτ σ −1 w |Sb | |Sp | τ ∈Sf 

τ ∈Sf 

1  1  = τ tbw (Qp Tτ σ −1 ) = tbw ( τ Qp Tτ σ −1 ) = tbw (c(Φf  ,σ , p)). |Sp | |Sp | τ ∈Sf 

τ ∈Sf 

˜ m , proving PropoBy Lemma 18.14, c(Φf  ,σ , p) and hence tbw (c(Φf  ,σ , p)) lies in K sition 18.5 and thus finishing the proof of Theorem 18.4. 18.3.3. The Theorem over Z. We claim now that the polynomials Z[xki,j ] on n-tuples of m × m matrices invariant under conjugation by GL(m, Z) which by §15.2.1 are generated over Z by the elements σi (M ), M ∈ W0 are also presented ˜ m. by the relations of the corresponding ideal K First, since the generators and relations defining the invariants for every field ˜ m ⊗ F equals F have integer coefficients this implies that the presentation S ⊗ F/K the ring of invariants for F any field finite or infinite. Moreover by Corollary 10.4, in each degree these algebras have the same dimension. This implies that ˜m ⊗ Q = K ˜ m and the claim follows. S∩K 18.3.4. The non commutative Theorem. We finally want to give the relations for the algebra of equivariant maps, as stated in Theorem 1.14 2). Theorem 18.17. The ideal Lm of defining relations for the non commutative algebra of equivariant maps from n tuples of m × m matrices to m × m matrices is generated by the elements σk (M ), k > m, by CHk (M ), k ≥ m and by their polarized forms. Proof. Let L˜m denote the ideal generated by the elements σk (M ), k > m, by CHk (M ), k ≥ m and by their polarized forms. Let a ∈ Lm . We need to show that a ∈ L˜m . Using an extra variable y we have that tr(ay) = σ1 (ay) is a relation for ˜ m generated by the elements σk (M ), k > m the invariants hence it is in the ideal K and their polarized forms  (100) σh1 ,...,hr (M1 , . . . , Mr ), hi > m. The element σ1 (ay) is of degree 1 in y so by homogeneity we can write it as a linear combination  σs1 ,...,sr ,1 (M1 , . . . , Mr , y)Cs σ1 (ay) = s=(s1 ,...sr )

˜ m or k =  si ≥ m. with either Cs ∈ K Let us now remark that σs1 ,...,s r ,1 (M1 , . . . , Mr , y), is obtained as a polarization with respect to the variable x → rj=1 tj xj of σk,1 (x, y) = (−1)k σ1 (CHk (x)y), by Lemma 4.19, and then by substituting xj with Mj . Thus σs1 ,...,sr ,1 (M1 , . . . , Mr , y) = (−1)k σ1 (bs (M1 , . . . , Mr )y) and if k > m, bs (M1 , . . . , Mr ) ∈ L˜m , It follows that  a = s bs Cs with either Cs ∈ Km or bs (M1 , . . . , Mr ) ∈ L˜m . This is our claim. 

˜m 18. Km VERSUS K

129

Remark 18.18. We have the symbolic identity CHm+1 (x) = CHm (x)x + (−1)m+1 σm+1 (x) The identity CHm (x) implies that CHm+1 (x) = (−1)m+1 σm+1 (x) so the vanishing of one of the two terms implies the vanishing of the other.

Part VI

The Schur algebra of a free algebra

132

VI. THE SCHUR ALGEBRA OF A FREE ALGEBRA

19. Preliminary facts Our goal in what follows is to apply the method of Example 2.11 to several matrices where the restriction map to diagonal matrices is not available. For this we shall need to perform a precise analysis of the Schur Algebras of the free algebras. Consider the free algebra F x1 , . . . , xn , and the algebra Sm [ξ1 , . . . , ξn ]GL(m) of invariant polynomial function on n, m × m matrices. where we denote by ξi the i-th generic matrix. Let us now recall the statement of Theorem 1.13: Theorem 19.1. Let F be any field or the integers. Given m ∈ N, for every n>0 (1) The map: det : F x1 , . . . , xn  → Sm [ξ1 , . . . , ξn ]GL(m) ,

f (x1 , . . . , xn ) → det(f (ξ1 , . . . , ξn ))

is multiplicative. (2) The induced map det : (F x1 , . . . , xn ⊗m )Sm → Sm [ξ1 , . . . , ξn ]GL(m) is surjective. (3) Its kernel is the ideal generated by all commutators. We will see that, the fact that the map det : (F x1 , . . . , xn ⊗m )Sm → Sm [ξ1 , . . . , ξn ]GL(m,F ) is surjective is a reformulation, due to Vaccarino, of the theorem of Donkin 1.10, while the fact that its kernel is the ideal generated by all commutators is a reformulation, due to Ziplies, of the theorem of Zubkov 1.12. := {ai , i ∈ I}, Let R be an algebra. Given two finite lists of elements of R, a  b := {bj , j ∈ J} and two functions α : I → N, β : J → N with d = i αi = j βj we now give a multiplication rule for the elements aα , bβ ∈ S d (R) = (R⊗d )Sd (defined in Remark 2.5 by Formula (11)). two disjoint sets Denote by a ◦ b := {ai bj , (i, j) ∈ I × J}. Take   of variables ti , i ∈ I; sj , j ∈ J and consider the two elements ( i∈I ti ai )⊗d , ( j∈J sj bj )⊗d . Formula (11) gives     ( ti ai )⊗d = tα a α , ( sj bj )⊗d = sβ bβ , α:I→N, |α|=d

i∈I

(



ti ai )⊗d (

i∈I



sj bj )⊗d = (

j∈J

α : I → N, |α| = d, β : J → N, |β| = d

Let us define M (α, β) := {γ : I × J → N |



ti sj ai bj )⊗d .

(i,j)∈I×J

We deduce, setting ts = {ti sj }:  tα sβ aα bβ =

(101)

β:J→N, |β|=d

j∈J



(ts)γ (a ◦ b)γ .

γ:I×J→N, |γ|=d

 i∈I

γi,j = βj ,

 j∈J

γi,j = αi }.

 In particular for such a γ we have i∈I, j∈J γi,j = d. Then, equating coefficients, we obtain the multiplication rule:

19. PRELIMINARY FACTS

Proposition 19.2. (102)

aα bβ =



133

(a ◦ b)γ .

γ∈M (α,β)

If R is an algebra with 1 we shall find it convenient to introduce a special no tation for the elements (a )α , with a := (1, a1 , . . . , ak ) and, under the assumption k k

i=1 αi ≤ d where α := (d − i=1 αi , α1 , . . . , αk ): (103)



(a )α := σd;α1 ,...,αk (a1 , . . . , ak ).

Remark 19.3. Notice that the symbol σd;α1 ,...,αk (a1 , . . . , ak ) is invariant under simultaneous permutation of the indices 1, . . . , k in the two lists (α1 , . . . , αk ), and (a1 , . . . , ak ). For the applications of the next section it is useful to cast Formula (102) in the special case, and with the notations, of Formula (103). That is, we choose a = (1, a1 , . . . , ah ), b = (1, b1 , . . . , bk ) so that a◦b is displayed by a sequence with entries (1, a1 , . . . , ah , b1 , . . . , bk , a1 b1 , . . . , ai bj , . . . , ah bk ). To be coherent we index the elements γi,j with I = {0, 1, . . . , h} and J = {0, 1, . . . , k}. It is then clear that  (104) σd;α1 ,...,αh (a1 , . . . , ah )σd;β1 ,...,βk (b1 , . . . , bk ) = σd,γ ([a ◦ b]γ ) γ∈M (α,β)

where we denote by [a ◦ b]γ the subsequence of (a1 , . . . , ah , b1 , . . . , bk , a1 b1 , . . . , ai bj , . . . , ah bk ) where the corresponding exponent γi,j = 0 and where by abuse of notation we still denote by γ its restriction to [a ◦ b]γ . Set, given α = (α1 , . . . , αh ), β = (β1 , . . . , βk )   N (α, β) = {γ : I ×J \{(0, 0)} → N | for j ≥ 1, γi,j = βj , for i ≥ 1, γi,j = αi }. i∈I

k

h

j∈J

It is useful to notice then that, assuming i=1 αi ≤ d, j=1 βj ≤ d and setting h k



α := (d− i=1 αi , α1 , . . . , αh ), β := (d− j=1 βi , β1 , . . . , βk ) we have a restriction

map from M (α , β ) to N (α, β). This map is injective since for γi,j ∈ M (α , β ) one has γ0,0 = d − (i,j)=(0,0) γi,j .   Lemma 19.4. 1) As soon as d ≥ hi=1 αi + kj=1 βj the restriction map gives a bijection between M (α , β ) and N (α, β). 2) Furthermore the right hand side of (104) equals σd;α1 ,...,αh ,β1 ,...,βk (a1 , . . . , ah , b1 , . . . , bk ) plus a sum of terms σd,γ ([a ◦ b]γ ) with the property that  (i,j)=(0,0)

γi,j
d is an ideal Id and the algebra S d X is naturally isomorphic to the quotient SA X/Id . (4) When X = {x} has a single element we have that SA x is the ring of symmetric polynomials in an arbitrary number of variables with coefficients in A. Proof. For each d ≥ 0, let δd : S˜A X → S d X be the algebra homomorphism such that πd ◦ δd+1 = δd . By definition it follows that S˜A X is a free module with basis given by the elements η f such that (d) ef if f ∈ Fd f δd (η ) = . 0 if f ∈ F \ Fd So setting δ(ef ) = η f , we get a linear isomorphism δ : SA X → S˜A X. Remark that by the definition the composition δd ◦ δ is compatible with the product. It follow that δ is also compatible with the product and the first point follows. After this, the proof of the remaining points is clear.  Recall that SX is a graded algebra with deg(ef ) = δ(f ), cf. (109). If X is finite, a simple counting argument gives (113)

∞  n=0

dim(SXn )tn =

∞ 

1 . (1 − td )|X|d d=1

138

6. THE SCHUR ALGEBRA OF A FREE ALGEBRA

Remark 20.9. Observe that in the algebra SX we have substitutional rules, that is every map φ : X → Z+ X induces a sequence of compatible homomorphisms S d (φ) : S d (F X) → S d (F X) for all d, and hence a homomorphism S(φ) : S(F X) → S(F X). In particular if for each j, φ(xj ) is a monomial, then φ(M ) is a monomial for each monomial M . This suggests the following notation. Take f ∈ F and {M1 , . . . , Mh } be its support, for each i = 1, . . . , h, set fi = f (Mi ). We denote ef by σf1 ,...,fh (M1 , . . . , Mh ) (1 = σ0 ). We then set S(φ)(σf1 ,...,fh (M1 , . . . , Mh )) = σf1 ,...,fh (φ(M1 ), . . . , φ(Mh )). Notice that, in view of this, it makes sense to consider σf1 ,...,fh (M1 , . . . , Mh ) also when the monomials (M1 , . . . , Mh ) are not distinct. What one obtains is an integer multiple of the element ef where f is defined as f (M ) = i |M =Mi fi . For example σ1,1,1 (x, x, x) = 6σ3 (x); σ2,1 (x, x) = 3σ3 (x). 20.1.3. Generators for SX, S d (ZX). Part 2) of Lemma 19.4 suggests to define a canonical filtration of the algebra SX. Definition 20.10. Set SXi := {σf1 ,...,fh (M1 , . . . , Mh ) |



fj ≤ i}.

j

Since for any i and j, SXi SXj ⊂ SXi+j by Lemma 19.4, we can take the associated graded algebra GrSX. Let us denote by W+ (X) the set of positive (M1 , . . . , Mh ) the class of degree monomials in the alphabet X. Denote by σ ¯f 1 ,...,fh σf1 ,...,fh (M1 , . . . , Mh ) in GrSX (whose degree is i fi ). We then have: Proposition 20.11. 1) The classes σ f1 ,...,fh (M1 , . . . , Mh ) with Mi distinct monomials, form a basis of GrSX. 2) Given monomials M1 , . . . Mh in W+ (X), one has σ f1 ,...,fh (M1 , . . . , Mh ) =

h 

σ fi (Mi ).

i=1

3) The algebra GrSX is a commutative ring generated by the classes σ i (M ) of the elements σi (M ), i > 0, M ∈ W+ (X) which multiply as divided powers that is

 i+j σ i (M )σ j (M ) = σ i+j (M ). i Proof. The first part follows directly from Lemma 19.4 and the fact that the elements σf1 ,...,fh (M1 , . . . , Mh ) with Mi distinct monomials, form a basis of SX (recall their symmetry by Remark 19.3). Furthermore by 2) of Lemma 19.4 σ f1 ,...,fh (M1 , . . . , Mh ) · σ fh+1 (Mh+1 ) = σ f1 ,...,fh ,fh+1 (M1 , . . . , Mh , Mh+1 ) which immediately implies 2) by induction. In particular σ i (M )σ j (N ) = σ i,j (M, N ) = σ j,i (N, M ) = σ j (N )σ i (M )

20. THE SCHUR ALGEBRA OF THE FREE ALGEBRA

139

so that the elements σ i (M ) commute and generate GrSX. Finally by Formula (12) we have

 i+j σ i,j (M, M ) = σ i+j (M ) i proving our proposition.  Given f ∈ F we want to exhibit an algorithm which expresses ef ∈ SX as a polynomial with integer coefficients in the elements σq (M ). Lemma 20.12. Given ai ∈ Z+ X, each element σi1 ,...,ik (a1 , . . . , ak ) is a polynomial with integer coefficients in the elements σq (M ) where M is some monomial in the ai and with q ≤ m := max(i1 , . . . , ik ). k Proof. We work by double induction on k and of the sum t := j=1 ij . If  k = 1 the element is σi1 (a1 ) and it is already of the given form. If kj=1 ij = 0, the element is 1 and there is nothing to prove. Otherwise compute the product σi1 (a1 )σi2 ,...,ik (a2 , . . . , ak ), and apply Formula (107). Thus, by Remark 19.5, we get the term σi1 ,...,ik (a1 , . . . , ak ) plus terms σγ with t < t and finish by induction.  In fact the formula is universal in the sense that we may take the formula when ai is the variable xi and obtain the result by substitution of xi with any ai . Theorem 20.13. 1) The algebra SX is generated by the elements σq (M ) as M runs over all primitive monomials Wp (X) (cf. 1.6). 2) In characteristic 0 the algebra SQ X is generated by the elements σ1 (M ) with M ∈ W+ (X). Proof. 1). Lemma 20.12 implies that SX is generated by the elements σq (M ), M ∈ W+ (X). Now if M = N k is not primitive we can apply the theory of commutative symmetric functions which gives the element σq (xk ) as a polynomial with integer coefficients in the elements σj (x), see page 4. 2). In characteristic 0 in the ring of symmetric functions σq (M ) corresponds to the elementary symmetric functions and σ1 (M k ) to the power sums ψk . So the statement follows from Formula (21).  Example 20.14. Let us develop as example the formula for the element in Formula (111) ef = a ⊗ a ⊗ ab + a ⊗ ab ⊗ a + ab ⊗ a ⊗ a = σ2,1 (a, ab). σ1 (ab) = 1 ⊗ 1 ⊗ ab + 1 ⊗ ab ⊗ 1 + ab ⊗ 1 ⊗ 1 σ2 (a) = a ⊗ a ⊗ 1 + a ⊗ 1 ⊗ a + 1 ⊗ a ⊗ a e = σ1 (ab)σ2 (a)−σ1 (aba)σ1 (a)+σ1 (aba2 ) = σ2 (a)σ1 (ba)−σ1 (a2 b)σ1 (a)+σ1 (a2 ba). Remark that the elements σi (p) are not free generators but satisfy complicated relations which have not been fully investigated. f

Corollary 20.15. The algebra SQ X is the universal enveloping algebra of F+ X considered as a Lie algebra. Proof. Consider the map σ1 : F+ X → SQ X From Formula (108) we deduce the identity [σ1 (a), σ1 (b)] = σ1 ([a, b]), ∀a, b ∈ F+ X. Therefore we deduce a algebra homomorphism from the universal enveloping algebra of F+ X to SQ X which by Theorem 20.13.2. is surjective.

140

6. THE SCHUR ALGEBRA OF A FREE ALGEBRA

In order to see that our homomorphism is also injective, we can assume that X is finite and then pass to the limit. The two algebras are graded and, if X is finite, they have the same graded dimension by Formula (113). This gives our claim.  Let us totally order X. We deduce a total ordering on the set Wp (X) of primitive monomials by setting p ≤ p if deg(p) ≤ deg(p ) and in case their degrees are equal p is lexicographically smaller than p . Identify the basis of monomials in F+ X with Wp (X) × N, mapping (p, r) to pr (r > 0) and totally order this basis using the lexicographic order on Wp (X) × N. Write the Lie algebra F+ X as F+ X = ⊕p∈Wp (X) Fp where Fp is the abelian subalgebra spanned by the elements pk , k > 0. Take the algebra Sp generated over Z by the elements σi (p). By Theorem 20.13 this algebra apart from the basis of monomials in σi (p) has also the basis formed by the elements 1 and ts=1 σis (pks ), i1 , . . . , it > 0, 0 < k1 < · · · < kt . Tensoring with Q we get the algebra Sp (Q) := Q ⊗ Sp which can be identified with the enveloping algebra of the abelian subalgebra Fp and hence by Corollary 20.15 is also the polynomial algebra in the generators σ1 (pk ). The PBW theorem, then gives Proposition 20.16.  The algebra SQ X is isomorphic, as vector space, to the ordered tensor product p∈W Sp (Q). Notice that using our various base for Sp we get bases for SQ . We now want to understand what happens over Z. Of course we get an injective linear map  P : Sp → SX. p∈Wp (X)

Theorem 20.17. The linear map P is an isomorphism. Proof. The filtration introduced in Definition 20.10 induces a filtration on Sp for every p ∈ Wp (X) and the map P is compatible with the filtrations. We deduce a map  GrP : GrSp → GrSX p∈Wp (X)

and, by a standard argument, it will sufficient to show that grP is surjective. Indeed if GrP is surjective take a ∈ SX. Let d be the minimum such  that a ∈ SXd . If d = 0 there is nothing to show. If d > 0, there exists a b ∈ ( p∈Wp (X) Sp )d such that P (b) − a ∈ SXd−1 and we are done by induction. By Proposition 20.11 the algebra GrSp is generated by the elements σ i (M ) M ∈ W+ (X), i > 0. If we write M = pk , p primitive we get that σ i (M ) = σ i (pk ) ∈ GrSp , so that σ i (M ) lies in the image of GrP . Remark now that GrP is a homomorphism of algebras so that its image is a subalgebra which, since it contains a set of generators of SX, must coincide with SX.  20.1.4. A universal commutative algebra. Given a commutative ring F and let R be a F -algebra which is a free F -module of finite or countable rank. Definition 20.18. The algebra by AdF (R) is defined as the maximal commutative quotient of SFd (R), that is the quotient modulo the ideal generated by commutators.

20. THE SCHUR ALGEBRA OF THE FREE ALGEBRA

141

When F = Z we just write Ad (R). From the definition we get Proposition 20.19. The mapping aR : R → S d (R) → AdF (R) is a multiplicative polynomial map homogenous of degree d which is universal with respect to multiplicative polynomial maps of homogeneous of degree d, R → U whose target U is a commutative algebra. If φ : R1 → R2 is a homomorphism of algebras, we deduce an induced homomorphism Φ : AdF (R1 ) → AdF (R2 ) which is surjective if φ is surjective. Thus, in order to understand formal computations in AdF (R) we may as well develop them in the case in which R is the free algebra F X. In fact we can even work with the limit algebra SF X of which SFd X is a quotient. We will denote the maximal commutative quotient of SF X by AF X and AZ X = AX. Proposition 20.20. In AX we have σh (ab) = σh (ba) for all h and all pairs of elements a, b. Proof. Let us first see that σ1 (ab) = σ1 (ba). We have σ1 (ab) = σ1 (a)σ1 (b) − σ1,1 (a, b),

σ1 (ba) = σ1 (b)σ1 (a) − σ1,1 (a, b)

so if we impose σ1 (a)σ1 (b) = σ1 (b)σ1 (a) we deduce σ1 (ab) = σ1 (ba). Observe that, by Theorem 20.13.2, this computation implies the full proposition for AF X if F contains Q. Let us now prove the statement over Z. We proceed by induction on h. We need a preliminary step. Let us assume that Proposition 20.20 holds for all r < h. Lemma 20.21. Given i1 , . . . , ik < h we have for all j1 , j2 , . . . , jk and q σi1 ,...,ik ,q (abj1 , . . . , abjk , b) = σi1 ,...,ik ,q (bj1 a, . . . , bjk a, b). in AX. Proof. We start to show that our claim holds for q = 0, that is, in AX σi1 ,...,ik (abj1 , . . . , abjk ) = σi1 ,...,ik (bj1 a, . . . , bjk a). By Lemma 20.12, σi1 ,...,ik (x1 , . . . , xk ) is a polynomial with integer coefficients in elements σr (M ) where r < h and the elements M are monomials in x1 , . . . , xk . If we make in any such monomial the substitution xs = abjs for all s or xs = bjs a for all s, we obviously get cyclically equivalent elements. Thus our claim follows by the inductive hypothesis. Suppose now q > 0 and that our Lemma holds for each q < q and for all k. Formula (107) then gives σi1 ,...,ik ,q (abj1 , . . . , abjk , b) = σi1 ,...,ik (abj1 , . . . , abjk )σq (b) − T with



T = 

σu,v,q (abj1 , . . . , abjk , abj1 +1 , . . . , abjk +1 , b)

uj +vj =ij , vj =q−q  >0

j

were (u, v) = (u1 , . . . , uk , v1 , . . . , vk ), Similarly we get σq,i1 ,...,ik (bj1 a, . . . , bjk a) = σq (b)σi1 ,...,ik (bj1 a, . . . , bjk a) − T

142

6. THE SCHUR ALGEBRA OF A FREE ALGEBRA

with T



T = 

σq ,u,v (b, bj1 a, . . . , bjk a, bj1 +1 a, . . . , bjk +1 a)

uj +vj =ij , vj =q−q  >0

j

Notice that we already know that σi1 ,...,ik (abj1 , . . . , abjk ) = σi1 ,...,ik (bj1 a, . . . , bjk a) in AX and by definition AX is commutative. It follows that our Lemma will follow once we show that T = T in AX. Take a summand σu,v,q (abj1 , . . . , abjk , abj1 +1 , . . . , abjk +1 , b) of T . We know that q < q so, by induction we deduce that this summand equals σu,v,q (bj1 a, . . . , bjk a, bj1 +1 a, . . . , bjk +1 a, b) which in turn equals σq ,u,v (b, bj1 a, . . . , bjk a, bj1 +1 a, . . . , bjk +1 a). We immediately deduce that T = T in AX since each summand of T equals one and exactly one summand of T .  Let us go back to the proof of Proposition 20.20. Formula (112) gives: σh (a)σh (b) = σh (ab) +

h−1 

σk,k,h−k (a, b, ab) + σh,h (a, b).

k=1

For each 0 < k < h, by Lemma 20.21 we get σk,k,h−k (a, b, ab) = σk,k,h−k (a, b, ba) in AX. This immediately implies our claim.  Recall that W0 denotes the set of Lyndon words in the alphabet X (see Definition 1.7). In Section 4.2.2 we have introduced the polynomial ring S = Z[σi (p)], i = 1, . . . , ∞, p ∈ W0 . Theorem 20.22. The algebra AX and the polynomial ring S are canonically isomorphic. Proof. By Proposition 20.20 we have that AX is generated over Z by the elements σi (p), i = 1, . . . , ∞, p ∈ W0 . So AX is canonically a quotient of S. The algebraic independence of the elements σi (p), i > 0, p ∈ W0 , in AX, will follow from their algebraic independence in AQ X, In turn by a change of variables, it is enough to prove that the elements σ1 (pk ), p ∈ W0 , k > 0, are algebraically independent. It is then enough to show that, for every d ∈ N once we pass to the quotient Ad (ZX) there are no relations of degree ≤ d. This follows by evaluating the algebra AQ X in the invariants of d×d matrices, where by Theorem 4.13 the relations among the elements σ1 (pk ) start only in degree d + 1.  Now we can recover in a different form the symbolic calculus which we started in §4.2.2. Given any homomorphism of the free algebra ZX in an algebra R this induces a homomorphism of Ad (ZX) in Ad (R). In particular a ring endomorphism of ZX is uniquely determined by any map xi → ai ∈ ZX. Passing to the limit we have thus a ring endomorphism of S = Z[σi (p)]. The image of an element σi (p) under this endomorphism is computed as follows. p maps

20. THE SCHUR ALGEBRA OF THE FREE ALGEBRA

143

to some polynomial in ZX, then we may first apply Formula (31) which gives the result as an expression in the elements σi (q k ) with q ∈ W0 and finally, for M = q k one uses the formulas for symmetric functions. Question. Is it possible to prove this without resorting to matrices? We finish this section, showing that the statement of Theorem 20.22 holds verbatim when the coefficient ring is a field. Theorem 20.23. If F is a field, the algebra AF X is the polynomial ring SF = F [σi (p)], i = 1, . . . , ∞, p ∈ W0 . Proof. Arguing as in Theorem 20.22 we only have to show that the elements σi (p), i = 1, . . . , ∞, p ∈ W0 are algebraically independent in AF X. We have seen that they are algebraically independent over Q so it is enough to prove that the dimension of AF X over F in each degree equals the dimension of AQ X over Q in the same degree. Now by Theorem 1.10 and Corollary 10.4, the algebra of invariants of n copies of m × m matrices has a dimension in each degree which is independent of the characteristic. It follows then that there are no algebraic relations of degree ≤ m in the generators, for all fields F . So since the algebra of invariants of n copies of  m × m matrices is a quotient of AF X for all m the claim follows. 20.1.5. The approach of Ziplies Vaccarino. We can finally prove the last Theorem 1.13. Theorem 20.24. F is a field or the integers, ξ1 , . . . , ξn generic m×m matrices: 1) The map det : F x1 , . . . , xn  → Sm [ξ1 , . . . , ξn ]GL(m) , f (x1 , . . . , xn ) → det(f (ξ1 , . . . , ξn )) is multiplicative and homogeneous of degree m. 2) The induced homomorphism det : (F x1 , . . . , xn ⊗m )Sm → Sm [ξ1 , . . . , ξn ]GL(m) is surjective 3) Ker (det) is the ideal generated by all commutators. Proof. 1) is clear. 2) follows already from the description of (F x1 , . . . , xn ⊗m )Sm given in Theorem 20.13 and Theorem 1.10. 3) Since Sm [ξ1 , . . . , ξn ]GL(m) is commutative, the map det factors through the ideal generated by commutators that is it gives a surjective map of Ad (F X) to the invariants. From Theorem 20.22 the algebra AX is the polynomial ring S = Z[σi (p)], and it maps to Ad (F X) in such a way that all the elements σi (a), i > d and their polarized forms are in the kernel by Formula (110). The composition det

S = AX → Ad (F X) −→ Sm [ξ1 , . . . , ξn ]GL(m) coincides with the evaluation of invariants. But all the defining relations for Sm [ξ1 , . . . , ξn ]GL(m) are given by Theorem 1.12 and by Formula (110) they are  already satisfied by Ad (F X), so the claim follows.

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General Index

Δ-filtration, 44 ∇-filtration, 44

full, 24 polynomial law, 9 projective cover, 40

adjacency, 52 affine scheme, 16 group, 16 algebra quasi hereditary, 46 Schur, 50 trace, 28 algebra with trace, 28

rational representation, 13 restitution, 24 rim of a diagram, 76 row bitableau, 59 Schur functors, 66 Schur module, 65 Schur symmetric function, 20 shape of the diagram, 51 simple degeneration, 103 skew diagram, 51 specialization, 24 Standard filtration, 68 straightening relation, 56

bitableaux semistandard, 60 canonical tableau, 53 Cayley–Hamilton identity, 30 composition, 95 cyclic equivalence, 5 dominance order, 51

trace formal, 28

equivariant maps, 26 essential extension, 40

weight, 51 of a bitableau, 63 dominant, 51 Weyl module, 69

free Schur algebra, 137 full polarization, 92 height, 20

Young diagram, 51 Young superclass, 103 Young tableau, 52 standard, 52

injective hull, 40 linear algebraic group, 12 matrix variable, 34 module socle, 42 superfluous, 40 tilting, 82 top, 42 monomial primitive, 4 multilinearization, 24 multiplicative map, 11 partial polarization, 92 polarization 149

Symbol Index

SA = SA X, 37 Tm , 38 Tst λ ⊂ Tλ , 88 A[G], 12 Dμ , 99 F+ X, 5 L ⊂ ∇λ , 73 Mλ , 88 R ⊂ F [xi,j ], 56 Rn , 50 SCg , 103 c Sh X, 110 Sm X, 108 Tμ , 99 Wp , 4 Y := Tβ , 80 Λ , 89 Θc , 99 Ξc , 103 π ¯Z,V , 111 ˇ 51 λ, λ t, 51 λ \ μ, 51 AdF (R), 140 AF X, 141 S, 37 S(c), 99 Fν , 80 Fd , 135 Fm X := Sm XGL(V ) , 108 P, 58 P(μ, s), 79 PA (M, N ), 9 SX, 137 S m (R), 11 SA X, 137 Y, 61 d Ac (F ), 97 Ac,d (A), 97 Tr(M ), 111 ∇λ (V ) , 20 πd , 136 σ1 = tr, 34

σi , 34 ˜ λ := Rm,t (λ), 72 Q εj1 ,...,jk , 9 |λ|, 51 bj , 125 e(λ), 104 eI = eα , 10 ef = σf1 ,...,fh (M1 , . . . , Mh ), 138 f ∗ , 40 ht(λ), 20 k : [1, . . . , N ] → [1, . . . , r], 95 o(λ), 104 pI , 54, 55 t(R) := {t(a), a ∈ R}, 28 W0 , 5 Tλ , 64 deg p, 99

151

Selected Published Titles in This Series 69 67 66 65

Corrado De Concini and Claudio Procesi, The Invariant Theory of Matrices, 2017 Sylvie Ruette, Chaos on the Interval, 2017 Robert Steinberg, Lectures on Chevalley Groups, 2016 Alexander M. Olevskii and Alexander Ulanovskii, Functions with Disconnected Spectrum, 2016

64 63 62 61

Larry Guth, Polynomial Methods in Combinatorics, 2016 Gon¸ calo Tabuada, Noncommutative Motives, 2015 H. Iwaniec, Lectures on the Riemann Zeta Function, 2014 Jacob P. Murre, Jan Nagel, and Chris A. M. Peters, Lectures on the Theory of Pure Motives, 2013

60 William H. Meeks III and Joaqu´ın P´ erez, A Survey on Classical Minimal Surface Theory, 2012 59 Sylvie Paycha, Regularised Integrals, Sums and Traces, 2012 58 Peter D. Lax and Lawrence Zalcman, Complex Proofs of Real Theorems, 2012 57 56 55 54

Frank Sottile, Real Solutions to Equations from Geometry, 2011 A. Ya. Helemskii, Quantum Functional Analysis, 2010 Oded Goldreich, A Primer on Pseudorandom Generators, 2010 John M. Mackay and Jeremy T. Tyson, Conformal Dimension, 2010

53 John W. Morgan and Frederick Tsz-Ho Fong, Ricci Flow and Geometrization of 3-Manifolds, 2010 52 Marian Aprodu and Jan Nagel, Koszul Cohomology and Algebraic Geometry, 2010 51 J. Ben Hough, Manjunath Krishnapur, Yuval Peres, and B´ alint Vir´ ag, Zeros of Gaussian Analytic Functions and Determinantal Point Processes, 2009 50 John T. Baldwin, Categoricity, 2009 49 J´ ozsef Beck, Inevitable Randomness in Discrete Mathematics, 2009 48 Achill Sch¨ urmann, Computational Geometry of Positive Definite Quadratic Forms, 2008 47 Ernst Kunz, David A. Cox, and Alicia Dickenstein, Residues and Duality for Projective Algebraic Varieties, 2008 46 Lorenzo Sadun, Topology of Tiling Spaces, 2008 45 Matthew Baker, Brian Conrad, Samit Dasgupta, Kiran S. Kedlaya, and Jeremy Teitelbaum, p-adic Geometry, 2008 44 Vladimir Kanovei, Borel Equivalence Relations, 2008 43 Giuseppe Zampieri, Complex Analysis and CR Geometry, 2008 42 Holger Brenner, J¨ urgen Herzog, and Orlando Villamayor, Three Lectures on Commutative Algebra, 2008 41 James Haglund, The q, t-Catalan Numbers and the Space of Diagonal Harmonics, 2008 40 Vladimir Pestov, Dynamics of Infinite-dimensional Groups, 2006 39 Oscar Zariski, The Moduli Problem for Plane Branches, 2006 38 37 36 35

Lars V. Ahlfors, Lectures on Quasiconformal Mappings, Second Edition, 2006 Alexander Polishchuk and Leonid Positselski, Quadratic Algebras, 2005 Matilde Marcolli, Arithmetic Noncommutative Geometry, 2005 Luca Capogna, Carlos E. Kenig, and Loredana Lanzani, Harmonic Measure, 2005

34 E. B. Dynkin, Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations, 2004 33 Kristian Seip, Interpolation and Sampling in Spaces of Analytic Functions, 2004 32 Paul B. Larson, The Stationary Tower, 2004

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/ulectseries/.

This book gives a unified, complete, and self-contained exposition of the main algebraic theorems of invariant theory for matrices in a characteristic free approach. More precisely, it contains the description of polynomial functions in several variables on the set of m × m matrices with coefficients in an infinite field or even the ring of integers, invariant under simultaneous conjugation. Following Hermann Weyl’s classical approach, the ring of invariants is described by formulating and proving • the first fundamental theorem that describes a set of generators in the ring of invariants, and • the second fundamental theorem that describes relations between these generators. The authors study both the case of matrices over a field of characteristic 0 and the case of matrices over a field of positive characteristic. While the case of characteristic 0 can be treated following a classical approach, the case of positive characteristic (developed by Donkin and Zubkov) is much harder. A presentation of this case requires the development of a collection of tools. These tools and their application to the study of invariants are exlained in an elementary, self-contained way in the book.

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