Graduate Texts in Mathematics 203 Editorial Board

S. Axler F.W. Gehring K.A. Ribet

Springer Science+ Business Media, LLC

Graduate Texts in Mathematics

T AKEUTI/ZARING. Introduction to 35 ALEXANDERIWERMER. Several Complex Axiomatic Set Theory. 2nd ed. Variables and Banach Algebras. 3rd ed.

2 OXTOBY. Measure and Category. 2nd ed. 36 KELLEYINAMIOKA et al. Linear 3 SCHAEFER. Topological Vector Spaces. Topological Spaces.

2nd ed. 37 MoNK. Mathematical Logic. 4 HILTON/STAMMBACH. A Course in 38 GRAUERT/FRITZSCHE. Several Complex

Homological Algebra. 2nd ed. Variables. 5 MAC LANE. Categories for the Working 39 ARVESON. An Invitation to C*-Algebras.

Mathematician. 2nd ed. 40 KEMENY/SNELL/KNAPP. Denumerable 6 HUGHES/PIPER. Projective Planes. Markov Chains. 2nd ed. 7 SERRE. A Course in Arithmetic. 41 APOSTOL. Modular Functions and Dirichlet 8 TAKEUTIIZARING. Axiomatic Set Theory. Series in Number Theory. 9 HUMPHREYS. Introduction to Lie Algebras 2nd ed.

and Representation Theory. 42 SERRE. Linear Representations of Finite 10 CoHEN. A Course in Simple Homotopy Groups.

Theory. 43 GILLMANIJERISON. Rings of Continuous II CONWAY. Functions of One Complex Functions.

Variable I. 2nd ed. 44 KENDIG. Elementary Algebraic Geometry. 12 BEALS. Advanced Mathematical Analysis. 45 LOEVE. Probability Theory !. 4th ed. 13 ANDERSON/FULLER. Rings and Categories 46 LOEVE. Probability Theory II. 4th ed.

of Modules. 2nd ed. 47 MOISE. Geometric Topology in 14 GoLUBITSKY/GuiLLEMIN. Stable Mappings Dimensions 2 and 3.

and Their Singularities. 48 SACHS/Wu. General Relativity for 15 BERBERIAN. Lectures in Functional Mathematicians.

Analysis and Operator Theory. 49 GRUENBERG/WEIR. Linear Geometry. 16 WINTER. The Structure of Fields. 2nd ed. 17 RosENBLATT. Random Processes. 2nd ed. 50 EDWARDS. Fermat's Last Theorem. 18 HALMOS. Measure Theory. 51 KLINGENBERG. A Course in Differential 19 HALMOS. A Hilbert Space Problem Book. Geometry.

2nd ed. 52 HARTSHORNE. Algebraic Geometry. 20 HUSEMOLLER. Fibre Bundles. 3rd ed. 53 MANIN. A Course in Mathematical Logic. 21 HUMPHREYS. Linear Algebraic Groups. 54 ORA VERIW ATKINS. Combinatorics with 22 BARNES/MACK. An Algebraic Introduction Emphasis on the Theory of Graphs.

to Mathematical Logic. 55 BROWN/PEARCY. Introduction to Operator 23 GREUB. Linear Algebra. 4th ed. Theory I: Elements of Functional 24 HOLMES. Geometric Functional Analysis Analysis.

and Its Applications. 56 MASSEY. Algebraic Topology: An 25 HEWITT/STROMBERG. Real and Abstract Introduction.

Analysis. 57 CROWELL/Fox. Introduction to Knot 26 MANES. Algebraic Theories. Theory. 27 KELLEY. General Topology. 58 KoBLITZ. p-adic Numbers, p-adic Analysis, 28 ZARISKIISAMUEL. Commutative Algebra. and Zeta-Functions. 2nd ed.

Vol.!. 59 LANG. Cyclotomic Fields. 29 ZARISKIISAMUEL. Commutative Algebra. 60 ARNOLD. Mathematical Methods in

Vol.!!. Classical Mechanics. 2nd ed. 30 JACOBSON. Lectures in Abstract Algebra I. 61 WHITEHEAD. Elements of Homotopy

Basic Concepts. Theory. 31 JACOBSON. Lectures in Abstract Algebra II. 62 KARGAPOLOV/MERLZJAKOV. Fundamentals

Linear Algebra. of the Theory of Groups. 32 JACOBSON. Lectures in Abstract Algebra 63 BOLLOBAS. Graph Theory.

Ill. Theory of Fields and Galois Theory. 64 EDWARDS. Fourier Series. Vol. I. 2nd ed. 33 HIRSCH. Differential Topology. 65 WELLS. Differential Analysis on Complex 34 SPITZER. Principles of Random Walk. Manifolds. 2nd ed.

2nd ed. (continued after index)

Bruce E. Sagan

The Symmetric Group Representations, Combinatorial Algorithms, and Symmetric Functions

Second Edition

With 31 Figures

Springer

Bruce E. Sagan Department of Mathematics Michigan State University East Lansing, MI 48824-1027 USA

Editorial Board

S. Axler Mathematics Department San Francisco State

University San Francisco, CA 94132 USA

F.W. Gehring Mathematics Department East Hall University of Michigan Ann Arbor, MI 48109 USA

K.A. Ribet Mathematics Department University of California

at Berkeley Berkeley, CA 94720-3840 USA

Mathematics Subject Classification (2000): 20Cxx, 20C30, 20C32, 05C85

Library of Congress Cataloging-in-Publication Data Sagan, Bruce Eli.

The symmetric group : representations, combinatorial algorithms, and symmetric functions I Bruce E. Sagan.

p. em. - (Graduate texts in mathematics ; 203) Includes bibliographical references and index. ISBN 978-1-4419-2869-6 ISBN 978-1-4757-6804-6 (eBook) DOI 10.1007/978-1-4757-6804-6

I. Representations of groups. 2. Symmetric functions. I. Title. II. Series. QA171 .S24 2000 512 .. 2-dc21 00-040042

Printed on acid-free paper.

First edition© 1991 by Wadsworth, Inc., Belmont, California.

© 2001 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 2001 Softcover reprint of the hardcover 2nd edition 2001

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+ Business Media, LLC , except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of gl!neral descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Production managed by Timothy Taylor; manufacturing supervised by Jeffrey Taub. Photocomposed copy prepared from the author's LaTeX files.

9 8 7 6 5 4 3 2 1

ISBN 978-1-4419-2869-6 SPIN 10770720

Preface to the 2nd Edition

I have been very gratified by the response to the first edition, which has resulted in it being sold out. This put some pressure on me to come out with a second edition and now, finally, here it is.

The original text has stayed much the same, the major change being in the treatment of the hook formula which is now based on the beautiful Novelli-Pak-Stoyanovskii bijection (NPS 97]. I have also added a chapter on applications of the material from the first edition. This includes Stanley's theory of differential posets (Stn 88, Stn 90] and Fomin's related concept of growths (Fom 86, Fom 94, Fom 95], which extends some of the combinatorics of Sn-representations. Next come a couple of sections showing how groups acting on posets give rise to interesting representations that can be used to prove unimodality results (Stn 82]. Finally, we discuss Stanley's symmetric function analogue of the chromatic polynomial of a graph (Stn 95, Stn ta].

I would like to thank all the people, too numerous to mention, who pointed out typos in the first edition. My computer has been severely reprimanded for making them. Thanks also go to Christian Krattenthaler, Tom Roby, and Richard Stanley, all of whom read portions of the new material and gave me their comments. Finally, I would like to give my heartfelt thanks to my editor at Springer, Ina Lindemann, who has been very supportive and helpful through various difficult times.

Ann Arbor, Michigan, 2000

v

Preface to the 1st Edition

In recent years there has been a resurgence of interest in representations of symmetric groups (as well as other Coxeter groups). This topic can be approached from three directions: by applying results from the general theory of group representations, by employing combinatorial techniques, or by using symmetric functions. The fact that this area is the confluence of several strains of mathematics makes it an exciting one in which to study and work. By the same token, it is more difficult to master.

The purpose of this monograph is to bring together, for the first time under one cover, many of the important results in this field. To make the work accessible to the widest possible audience, a minimal amount of prior knowledge is assumed. The only prerequisites are a familiarity with elemen­tary group theory and linear algebra. All other results about representations, combinatorics, and symmetric functions are developed as they are needed. Hence this book could be read by a graduate student or even a very bright undergraduate. For researchers I have also included topics from recent journal articles and even material that has not yet been published.

Chapter 1 is an introduction to group representations, with special empha­sis on the methods of use when working with the symmetric groups. Because of space limitations, many important topics that are not germane to the rest of the development are not covered. These subjects can be found in any of the standard texts on representation theory.

In Chapter 2, the results from the previous chapter are applied to the symmetric group itself, and more highly specialized machinery is developed to handle this case. I have chosen to take the elegant approach afforded by the Specht modules rather than working with idempotents in the group algebra.

The third chapter focuses on combinatorics. It starts with the two famous formulae for the dimensions of the Specht modules: the Frame-Robinson­Thrall hook formula and the Frobenius-Young determinantal formula. The centerpiece is the Robinson-Schensted-Knuth algorithm, which allows us to describe some of the earlier theorems in purely combinatorial terms. A thor­ough discussion of Schiitzenberger's jeu de taquin and related matters is in­cluded.

Chapter 4 recasts much of the previous work in the language of symmet­ric functions. Schur functions are introduced, first combinatorially as the generating functions for semistandard tableaux and then in terms of sym­metric group characters. The chapter concludes with the famous Littlewood­Richardson and Murnaghan-Nakayama rules.

My debt to several other books will be evident. Much of Chapter 1 is based on Ledermann's exquisite text on group characters [Led 77]. Chapter

Vll

viii PREFACE TO THE 1ST EDITION

2 borrows heavily from the monograph of James [Jam 78], whereas Chapter 4 is inspired by Macdonald's already classic book [Mac 79]. Finally, the third chapter is a synthesis of material from the research literature.

There are numerous applications of representations of groups, and in par­ticular of the symmetric group, to other areas. For example, they arise in physics [Boe 70], probability and statistics [Dia 88], topological graph the­ory [Whi 84], and the theory of partially ordered sets [Stn 82]. However, to keep the length of this text reasonable, I have discussed only the connections with combinatorial algorithms.

This book grew out of a course that I taught while visiting the Universite du Quebec a Montreal during the fall of 1986. I would like to thank l'equipe de combinatoire for arranging my stay. I also presented this material in a class here at Michigan State University in the winter and spring of 1990. I thank my students in both courses for many helpful suggestions (and those at UQAM for tolerating my bad French). Francesco Brenti, Kathy Dempsey, Yoav Dvir, Kathy Jankoviak, and Scott Mathison have all pointed out ways in which the presentation could be improved. I also wish to express my appreciation of John Kimmel, Marlene Thorn, and Linda Loba at Wadsworth and Brooks/Cole for their help during the preparation of the manuscript. Because I typeset this document myself, all errors can be blamed on my computer.

East Lansing, Michigan, 1991

Contents

Preface to the 2nd Edition

Preface to the 1st Eition

1 Group Representations 1.1 Fundamental Concepts . . . . . . . 1.2 Matrix Representations . . . . . . 1.3 G-Modules and the Group Algebra 1.4 Reducibility ............ . 1.5 Complete Reducibility and Maschke's

Theorem ................ . 1.6 G-Homomorphisms and Schur's Lemma 1. 7 Commutant and Endomorphism Algebras 1.8 Group Characters ........... . 1.9 Inner Products of Characters . . . . . 1.10 Decomposition of the Group Algebra . 1.11 Tensor Products Again ........ . 1.12 Restricted and Induced Representations 1.13 Exercises

2 Representations of the Symmetric Group 2.1 Young Subgroups, Tableaux, and Tabloids 2.2 Dominance and Lexicographic Ordering 2.3 Specht Modules . . . . . . . . . . . . . 2.4 The Submodule Theorem ...... . 2.5 Standard Tableaux and a Basis for s>-. 2.6 Garnir Elements ........ . 2.7 Young's Natural Representation. 2.8 The Branching Rule ...... . 2.9 The Decomposition of !vJI-L ... . 2.10 The Semistandard Basis for Hom(S\ !vJI-L) 2.11 Kostka Numbers and Young's Rule 2 .12 Exercises . . . . . . . . .

ix

v

vii

1 1 4 6

10

13 18 23 30 33 40 43 45 48

53 53 57 60 63 66 70 74 76 78 82 85 86

X

3 Combinatorial Algorithms 3.1 The Robinson-Schensted Algorithm . 3.2 Column Insertion .......... . 3.3 Increasing and Decreasing Subsequences 3.4 The Knuth Relations . . . . . . . . 3.5 Subsequences Again . . . . . . . . 3.6 Viennot's Geometric Construction 3.7 Schiitzenberger's Jeu de Taquin 3.8 Dual Equivalence . 3.9 Evacuation ........ . 3.10 The Hook Formula .... . 3.11 The Determinantal Formula 3.12 Exercises ......... .

4 Symmetric Functions 4.1 Introduction to Generating Functions 4.2 The Hillman-Grass! Algorithm .. 4.3 The Ring of Symmetric Functions 4.4 Schur Functions . . . . . . . . . . . 4.5 The Jacobi-Trudi Determinants .. 4.6 Other Definitions of the Schur Function 4. 7 The Characteristic Map . . . . . 4.8 Knuth's Algorithm ....... . 4.9 The Littlewood-Richardson Rule 4.10 The Murnaghan-Nakayama Rule 4.11 Exercises

5 Applications and Generalizations 5.1 Young's Lattice and Differential Posets. 5.2 Growths and Local Rules 5.3 Groups Acting on Posets . . . . . 5.4 Unimodality ........... . 5.5 Chromatic Symmetric Functions 5.6 Exercises ............ .

Bibliography

Index

CONTENTS

91 91 95 97 99

102 106 112 116 121 124 132 133

141 142 147 151 155 158 163 167 169 174 179 185

191 191 197 204 208 213 218

223

230

List of Symbols Symbol Meaning Page

A poset 58 Ak kth rank of a graded poset 195 As rank selected poset 204

a" alternant corresponding to composition J-t 164 a\a1 composition a with first part a 1 removed 180

(3 set partition 215

c complex numbers 4 chn characteristic map 167 Cn cyclic group of order n 5 Ct column-stabilizer group of Young tableau t 60 Com X commutant algebra of representation X 23 C[G] group algebra of G over C 8 C[S] vector space generated by set S over C 7 C[[x]] ring of formal power series in x over C 142 Cx column insertion operator for integer x 95

>. CIJ.V Littlewood-Richardson coefficient 175 x(g) character of group element g 30 XK character value on conjugacy class K 32 xdef defining character of Sn 31 X reg regular character 31 x-1-~ restriction of character x from G to H 45 xt~ induction of character X from h to G 45

D down operator of a poset 196 dk(7r) length of 1r's longest k-decreasing subsequence 103 6.Q delta operator applied to tableau Q 121 degX degree of representation X 4 DesP the descent set of a tableau P 206

€>. Ath elementary symmetric function 154 en nth elementary symmetric function 152 €t polytabloid associated with Young tableau t 61 E(r) edges of graph r 213

xi

xii LIST OF SYMBOLS

Symbol Meaning Page

EndV endomorphism algebra of module V 23 evQ the evacuation tableau of Q 122 € identity element of a group 1

fs weight generating function of weighted set S 145 !>. number of standard A-tableaux 73 ¢>. character of permutation module lvf >. 56

G group 1 9A,B Garnir element of a pair of sets A, B 70 GLd d x d complex general linear group 4 GP generalized permutations 169 GP' generalized permutations, no repeated columns 172 r graph 213

hn nth complete homogeneous symmetric function 152 h>. Ath complete homogeneous symmetric function 154 Hom(V, W) all module homomorphisms from V to W 22 h·. t,J hooklength of cell ( i, j) 124 H-· t,) hook of cell ( i, j) 124 hv hooklength of cell v 124 Hv hook of cell v 124

I identity matrix 23 Id d x d identity matrix 24 ik ( 7r) length of 1r's longest k-increasing subsequence 103 i>.(V) number of stable partitions of V of type A 215 imB image of B 21

j(P) jeu de taquin tableau of P 116 Jc(P) forward slide on tableau P into cell c 113 f(P) backward slide on tableau P into cell c 113 jq(P) sequence of forward slides on P into Q 118 jp(Q) sequence of backward slides on Q into P 118

kerB kernel of B 21 Kg conjugacy class of a group element g 3 k>. size of K>. 3 K>. conjugacy class in Sn corresponding to A 3 K>.J.L Kostka number 85 /'£t signed column sum for Young tableau t 61

LIST OF SYMBOLS xiii

Symbol Meaning Page

l (.A) length (number of parts) of partition ).. 152 ll(~) leg length of rim hook ~ 180 ).. integer partition 2

integer composition 67 )..' conjugate or transpose of partition ).. 103 ( )..1' )..2' ... ' .At) partition given as a sequence 2

composition given as a sequence 15 .A/fl skew shape 112 .A\~ partition ).. with rim hook~ removed 180 .A(/3) type of the set partition f3 215 A algebra of symmetric functions 151 A1 symmetric functions in l variables 163 An symmetric functions homogeneous of degree n 152

Matd full d x d matrix algebra 4 m.x .Ath monomial symmetric function 151 !vJA permutation module corresponding to ).. 56

p partial tableau 92 p(n) number of partitions of n 143 Pn nth power sum symmetric function 152 P.>. .Ath power sum symmetric function 154 P(7r) P-tableau of permutation 1r 93 Pr chromatic polynomial of graph r 214 lP' positive integers 214 7r permutation 1 1rp row word of the Young tableau P 101 1rr reversal of permutation 1r 97

* top row of generalized permutation 1r 169 if' bottom row of generalized permutation 1r 169

R-S ( ) 7r f---+ P, Q Robinson-Schensted map 92

7r R~K (P,Q) Robinson-Schensted-Knuth map 170

7r R~=~F (P, Q) dual Robinson-Schensted-Knuth map 172

Q(7r) Q-tableau of permutation 1r 93

R(G) vector space of class functions on G 32 Rn class functions of Sn 167 Rt row-stabilizer group of Young tableau t 60 rx row-insertion operator for the integer x 93 rk the rank function in a poset 195

xiv LIST OF SYMBOLS

Symbol Meaning Page

SA group of permutations of A 20 8>. Schur function associated with >. 155 S>.fp. skew Schur function associated with .A/ J.l 175 S>. Young subgroup associated with partition >. 54 s>- Specht module associated with partition >. 62 sgn(rr) sign of permutation rr 4 sht shape of Young tableau t 55 Sn symmetric group on { 1, 2, ... , n} 1

t Young tableau 55 T generalized Young tableau 78 {t} row tabloid of t 55 [t] column tabloid oft 72 7>.1-1 generalized tableaux, shape A, content J.l 78

~~-~ semistandard tableaux, shape >., content J.l 81 0 homomorphism of modules 18 Or homomorphism corresponding to tableau T 80 Or restriction of Or to a Specht module 81

u up operator of a poset 196

VQ(P) vacating tableau for ]Q(P) 118 vP(Q) vacating tableau for jP(Q) 118 v(r) vertices of graph r 213

w.L orthogonal complement of W 15 wt weight function 145

X the set ofvariables {x1,x2,xJ, ... } 151 X(g) matrix of g in the representation X 4 xr monomial weight of a tableau T 155 x~-' monomial weight of a composition J.l 155 X.J,.~ restriction of representation X from G to H 45 Xt~ induction of representation X from H toG 45 Xr chromatic symmetric function of graph r 214 ~ rim or skew hook 180

y Young's lattice 192

ZA center of algebra A 27 Zg centralizer of group element g 3 Z>. size of Z 9 where g E Bn has cycle type >. 3 z integers 204

LIST OF SYMBOLS XV

Symbol Meaning Page

6 minimum of a poset 195 i maximum of a poset 204 1a trivial representation of G 5 (1m1,2m2, ... ) partition given by multiplicities 2 C>! equivalence of modules 19

slide equivalence of tableaux 114 * C>! dual equivalence of tableaux 117 K C>! Knuth equivalence 100 K* C>! dual Knuth equivalence 111 p C>! P-equivalence 99 Q C>! Q-equivalence 111 1

C>! Knuth relation of the first kind 99 1* C>! dual Knuth relation of the first kind 111 2

C>! Knuth relation of the second kind 99 2* C>! dual Knuth relation of the second kind 111 < subgroup relation 9

submodule relation 10 lexicographic order on partitions 59

-< covering relation in a poset 58 I> dominance order on partitions 58

dominance order on tabloids 68 f- is a partition of for integers 53

is a partition of for sets 215 EB direct sum of matrices 13

direct sum of vector spaces 13 181 tensor product of matrices 25

tensor product of representations 43 tensor product of vector spaces 26

(x,¢) inner product of characters x and¢ 34 x·¢ product of characters 168 AxB product of posets 198 GIH wreath product of groups 212 u union of tableaux 120 ~ disjoint union 45 1·1 cardinality of a set 3

sum of parts of a partition 54 1\ meet operation in a poset 192 v join operation in a poset 192