Littlewood-Richardson rules for ordinary and projective representations of symmetric groups

Table of contents :
Signature Page ................................................................................................. iii
Table of Contents ............................................................................................ iv
Acknowledgements .......................................................................................... vi
Vita and Publications .................................................................................... vii
Abstract .......................................................................................................... viii
Introduction ....................................................................................................... 1
I Chapter 1. Representations and the LR rule ............................................. 4
1.1 Representation theory of finite groups ........................................... 4
1.2 Representations of symmetric groups ........................................... 9
1.3 Projective representations ............................................................... 18
1.4 Projective representations of symmetric groups.........................24
II Chapter 2. Compatibility .............................................................................. 38
2.1 Definitions .......................................................................................... 38
2.2 Schensted insertion .......................................................................... 44
2.3 Standard Labelling ......................................................................... 55
2.4 The Slide Lemma and jeu de taquin ............................................. 63
2.5 Shuffles................................................................................................77
2.6 Insertion and jeu de taquin ............................................................ 80
2.7 Insertion symmetries ........................................................................ 84
2.8 Reading tableaux .............................................................................. 90
2.9 Compatibility .................................................................................... 93
2.10 Various compatibilities ................................................................. 99
III Chapter 3. Shift compatibility ................................................................... 102
3.1 Shifted Tableaux ............................................................................ 102
3.2 Shifted insertion ............................................................................ 105
3.3 Labelling and shifted insertion ................................................... 117
3.4 Shift compatibility ........................................................................ 122
3.5 Row words and shuffles ................................................................. 126
3.6 Nonskew shifted shapes and reading tableaux ........................ 133
3.7 Two row rectangles ........................................................................ 138
3.8 Involutions on generalized shifted tableaux ............................. 144
IV Chapter 4. Littlewood-Richardson rules and generalizations ...... 148
4.1 Introduction ..................................................................................... 148
4.2 The Kostka matrix and r-pairing .............................................. 157
4.3 The inverse Kostka matrix and special rim hook tabloids .. 166
4.4 The classical Littlewood-Richardson rule .................................. 172
4.5 Bijections ......................................................................................... 175
Conclusion ..................................................................................................... 180
Bibliography .................................................................................................. 181

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Order N um ber 9208201

L ittlew ood-R ichardson rules for ordinary and projective representations o f sym m etric groups Shimozono, Mark Masami, Ph.D. University of California, San Diego, 1991

UMI

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22 00643 7529 U N IV E R S IT Y OF C A L IF O R N IA , S A N D IE G O

L ittlew ood-R ichardson R ules for O rdinary and P rojective R ep resen tation s o f Sym m etric G roups

A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics

by

Mark Shimozono

Committee in charge: Professor Professor Professor Professor Professor

Jeffrey B. Remmel, Chair Edward A. Bender Adriano M. Garsia Michael Saks S. Gill Williamson

1991

cdA

11

The dissertation of Mark Shimozono is approved, and it is acceptable in quality and form for publication on microfilm:

Chair

University of California, San Diego 1991

iii

TABLE OF CO NTENTS

Signature P ag e................................................................................................. iii Table of C on ten ts............................................................................................ iv Acknowledgements.......................................................................................... vi Vita and Publications....................................................................................vii A b stract.......................................................................................................... viii Introduction....................................................................................................... 1 I Chapter 1. Representations and the LR rule .............................................4 1.1 Representation theory of finite groups...........................................4 1.2 Representations of symmetric groups ........................................... 9 1.3 Projective representations...............................................................18 1.4 Projective representations of symmetric groups.........................24 II Chapter 2. C om patibility..............................................................................38 2.1 Definitions..........................................................................................38 2.2 Schensted insertion.......................................................................... 44 2.3 Standard L abelling ......................................................................... 55 2.4 The Slide Lemma and jeu de taquin.............................................63 2.5 Shuffles................................................................................................77 2.6 Insertion and jeu de ta q u in ............................................................80 2.7 Insertion symm etries........................................................................84 2.8 Reading tab leau x ..............................................................................90 2.9 Com patibility.................................................................................... 93 2.10 Various com patibilities................................................................. 99 III Chapter 3. Shift com patibility................................................................... 102 3.1 Shifted T ableaux ............................................................................ 102 3.2 Shifted insertion ............................................................................ 105 3.3 Labelling and shifted insertion ...................................................117 3.4 Shift co m p atib ility ........................................................................ 122 3.5 Row words and shuffles................................................................. 126 3.6 Nonskew shifted shapes and reading tableaux ........................133 3.7 Two row rectangles........................................................................ 138 3.8 Involutions on generalized shifted tableaux .............................144

IV

Chapter 4. Littlewood-Richardson rules and generalizations......148 4.1 Introduction.....................................................................................148 4.2 The Kostka m atrix and r-pairing .............................................. 157 4.3 The inverse Kostka m atrix and special rim hook tabloids .. 166 4.4 The classical Littlewood-Richardson ru le .................................. 172 4.5 B ijections.........................................................................................175 C onclusion..................................................................................................... 180 B ibliography..................................................................................................181

ACKNOW LEDGEM ENTS

I would like to thank my family for their unwavering support in seasons of frustration and uncertainty as well as in times of victory. I am indebted to my instructors. Jeff Remmel had a project on hand when I needed one and was always available to hear both my imprecise babbling and reasonable ideas. The combinatorics group at UCSD has been warm, enthusiastic, and sup­ portive. I will miss the fellowship of APM 2250.

VITA

November 21, 1963

Born, Monterey, California, USA

1983

B. Sc. Mathematics, Biola University

1986

M. S. Mathematics, Stanford University

1985-90

Teaching Assistant Department of Mathematics University of California, San Diego

1991

Ph. D. Mathematics, University of California, San Diego

PU B L IC A T IO N S A Simple Proof o f the Littlewood-Richardson Rule and Variants. In preparation. A New Shifted Littlewood-Richardson Rule. In preparation.

A B S T R A C T OF T H E D ISSE R TA T IO N

Littlewood-Richardson Rules for Ordinary and Projective Representations of Symmetric Groups by Mark Shimozono Doctor of Philosophy in Mathematics University of California, San Diego, 1991 Professor Jeffrey B. Remmel, Chair

The main result of this thesis is a new shifted Littlewood-Richardson rule. In 1934 Littlewood and Richardson stated a combinatorial interpretation for the coefficients arising from the decomposition of the induction product of represen­ tations of symmetric groups. Much later D. W hite proved a theorem leading to an alternative description of these coefficients. These rules provide examples of the rich relationships between the representation theory of symmetric groups, symmetric functions, and tableau combinatorics. The theory of projective rep­ resentations of symmetric groups also has strong connections with symmetric functions and the shifted tableau theory of Sagan and Worley. Recently Stembridge gave the first analogue of the Littlewood-Richardson rule in the context of projective representations of symmetric groups. This rule computes the de­ composition of an induced product of projective representations of symmetric v iii

groups, where the multiplicities are expressed in terms of shifted tableaux. This thesis presents a new version of Stembridge’s shifted rule. This new shifted rule allows for the definition of an involution on generalized shifted tableaux which shows directly th at the definition of the skew Schur Q functions as the generat­ ing function of generalized shifted tableaux, yields a symmetric function. This involution is a shifted analogue of the “automorphisms of conjugation” given by Lascoux and Schiitzenberger. As in many of the proofs of the classical rule, the main tools employed are the Robinson-Schensted-Knuth correspondence and Schutzenberger’s jeu de taquin, together with their shifted analogues. Much of this theory is revisited, with particular emphasis on the connection of slides with recording tableaux of shuffles of Schensted row and column insertions. The last chapter gives a unified proof of the classical rule, its aforementioned alternative version, and their extensions. This proof uses a single directly defined involu­ tion. In the context of these extensions, the Littlewood-Richardson rule and the alternative rule are seen to be identical. This symmetry does not seem to extend to the shifted case.

Introd u ction The general topic of this thesis is the Littlewood-Richardson (LR) rule and several of its analogues, whose context is the theory of linear and projec­ tive representations of symmetric groups. The originators of the linear theory are Frobenius, who used symmetric functions to describe the irreducible characters, and Young, who employed tableaux to construct the irreducible representations. The projective theory is due to Schur [Schu], who also used symmetric functions to give the irreducible projective characters. A few decades later Littlewood and Richardson [LR] stated (without proof) the rule th at bears their names, a com­ binatorial recipe for decomposing the representation induced from an irreducible representation of a Young subgroup. Most of the early proofs of the LR rule used tableaux combinatorics. The first proof was given by Robinson [Rob]. Although incomplete, it introduced a version of the algorithm now known as the RobinsonSchensted-Knuth (RSK) correspondence. The first satisfactory proofs were given by Schutzenberger [Schu3] via his jeu de taquin, an algorithm equivalent to RSK, and Thomas [Tho2], using the RSK correspondence. There are versions of the LR rule can be derived from the work of D. W hite [Whi] and Hillman and Grassl [HG]; see Remmel-Whitney [RW]. These rules are more flexible than the LR rule. The tableaux combinatorics related to the projective representations of symmetric groups has only recently been developed by Sagan [Sag] and Worley [Wor]. Their shifted tableaux theory contains analogues of the RSK correspon­ dence and jeu de taquin. Nazarov [N] has constructed the irreducible projective representations of symmetric groups analogous to Young’s orthogonal represen­ tations. Stembridge [Stem] has elucidated the relationship between the theory of projective characters of symmetric groups and this shifted tableaux theory. Of par­ ticular interest to us is his projective analogue of induction from Young subgroups

1

2

and the first shifted analogue of the LR rule. The main result here is a shifted analogue of the result of Hillman-Grassl. This new shifted rule provides an alternative method for calculating the values given by Stembridge’s shifted LR rule. An immediate consequence is an invo­ lution on shifted tableaux which proves directly th at the definition of the (skew) Schur Q function as the generating function of shifted tableaux yields a symmetric function. This involution is the shifted analogue of Lascoux and Schiitzenberger’s “automorphisms of conjugation” [LS] rather than the involution of Bender-Knuth |BK], To this end, the first chapter briefly reviews the linear and projective representation theory of symmetric groups and (shifted) tableaux theory necessary to state the various combinatorial rules and indicate their representation-theoretic significance. The second chapter gives the non-shifted combinatorial preliminaries. It is mostly expository, but occasionally there are explicit statem ents of results which are off the beaten path. The RSK correspondence and jeu de taquin are revisited in light of a seemingly neglected corollary of a lemma of Schensted. This key lemma makes explicit the immediate connection between the slides of jeu de taquin and the recording tableaux of Schensted row and column insertion, a notion implicit in the work of Haiman [Hal]. This connection provides a remarkably concise proof th at the jeu de taquin is well-defined. It also elegantly reveals the properties of evacuation [Schu2], [Hal]. This development of insertion and jeu de taquin culminates in a proof of (the row insertion version of) the Remmel-Whitney (RW) rule. This proof is designed so th at many of its components can be converted to help prove the new shifted rule. Chapter 3 starts with the shifted combinatorial preliminaries such as shifted insertion, with emphasis on Stembridge’s hook words [Stem]. Then the main result is proven and the involution on shifted tableaux

is given. This is accomplished using the key lemma, which allows for a delicate analysis of the slides which deform the tableaux described by (the row insertion version of) the RW rule into the shifted tableaux counted by the new shifted rule. Chapter 4 is self-contained and gives unified, Schenstedless direct proofs of the LR rule, RW rule, and “generalizations” of both, using completely different methods. Employing only a simple involution, it avoids inclusion-exclusion and fancy applications of the involution principle. The LR and RW rules axe seen to be completely equivalent; this is only apparent upon examination of the versions of these rules which compute the inner product of a pair of skew Schur S functions.

C hapter 1. R ep resen tation s and th e LR rule

The purpose of this chapter is to introduce the LR rule and its analogues. The first section recounts the main results of the theory of representations of finite groups over C. The second section specializes this theory to the symmetric groups, whose representations are then connected to symmetric functions and tableaux. An induction product of representations of symmetric groups defines coefficients which can be calculated combinatorially by the LR and RW rules. The statements of these rules concludes the section. Section 3 gives an overview of the theory of projective representations of finite groups over C. Section 4 again specializes to the symmetric groups and connects their projective representations to symmetric functions and shifted tableaux. Stembridge’s rule and our rule are given, their representation-theoretic significance stemming from a projective analogue of the above induction product.

S ection 1.1. R ep resen tation th eory o f finite groups

This section is a brief overview of the well-understood theory of finite dimensional representations of a finite group G over the complex numbers C. The goal is to categorize the representations of G and give a few construc­ tions for representations. A representation A : G —> G L (F) of G is a group homomorphism from G to the group of linear automorphisms of a finite dimensional vector space V. Define the dimension of the representation A as the dimension of V. Extending A by linearity yields a C-algebra homomorphism A : C G —►End(V) from the group algebra C G to the algebra of linear transformations of V . A gives V a CG-module structure via g . v = A(g)v for all g 6 G, v € V.

4

Say th a t the representations A : G —* GL(V) and B : G —* GL(W ) are equivalent if there is a linear isomorphism T : V —►W such th at T A (g)v = B (g)T v for all g € G and v E V , i.e., T is a C G module isomorphism. The character \A of a representation A : G —►G L (F) is the map

xa

•

G —» C given by XA{g) = tr(A (g)), where tr is the trace. A character (or any function from G to C ) can be viewed as an element of C G by C-linear extension, i.e.

g£G

Now t r ^ S T S - 1) = tr(T ) for all T € E nd(F ) and S G GL(V). It follows that XA(hgh_1) = XA^g) for all g, h 6 G, i.e.,

xa

hes in the center Z (C G ) of CG.

Characters determine equivalence of representations. P r op o sitio n 1.1.1. Two representations o f G are equivalent i f and only i f they have the same character. Define the direct sum of the representations A : G —* GL(V) and B : G —> GL(W ) by A @ B : G —*■GL(V® W ) given by ( A®B) ( g ) ( v + w) = A (g)v-\-B(g)w for all g £ G, v G V , w 6 W . Its character satisfies XA@B(g) = X.a( C is given by 1c (g) = 1 if g € C and lc(