The Representation Theory of the Symmetric Group [Paperback ed.] 9780521104128, 9780521302364

Table of contents :
Title page
Contents
Editor's Statement
Foreword
Introduction
Preface
List of Symbols
1. Symmetric Groups and Their Young Subgroups
2. Ordinary Irreducible Representations and Characters of Symmetric and Alternating Groups
3. Ordinary Irreducible Matrix Representations of Symmetric Groups
4. Representations of Wreath Products
5. Applications to Combinatorics and Representation Theory
6. Modular Representations
7. Representation Theory of S_n over an Arbitrary Field
8. Representations of General Linear Groups
Appendix I. Tables
Appendix II. Notes and References
II.D. References
Index

Citation preview

The Representation Theory of the Symmetric Group

https://doi.org/10.1017/CBO9781107340732

GIAN-CARLO ROTA, Editor ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS

Volume

10

Section LUIS A. SANTALO Integral Geometry and Geometric Probability, 1976 (2nd printing, with revisions, 1979) GEORGE E. ANDREWS The Theory of Partitions, 1976 (2nd printing, 1981) ROBERT J. McELIECE The Theory of Information and Coding A Mathematical Framework for Communication, 1977 (2nd printing, with revisions, 1979) WILLARD MILLER, Jr. Symmetry and Separation of Variables, 1977 DAVID RUELLE Thermodynamic Formalism The Mathematical Structures of Classical Equilibrium Statistical Mechanics, 1978 HENRYK MINC Permanents, 1978 FRED S. ROBERTS Measurement Theory with Applications to Decisionmaking, Utility, and the Social Sciences, 1979 L. C. BIEDENHARN and J. D. LOUCK Angular Momentum in Quantum Physics: Theory and Application, 1981 L. C. BIEDENHARN and J. D. LOUCK The Racah-Wigner Algebra in Quantum Theory, 1981 JOHN D. DOLLARD and CHARLES N. FRIEDMAN Product Integration with Application to Differential Equations, 1979

Probability

Number Theory Probability

Special Functions Statistical Mechanics

Linear Algebra Mathematics and the Social Sciences Mathematics of Physics Mathematics of Physics Analysis

GIAN-CARLO ROTA, Editor ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS

11

12

13

14

15

16

WILLIAM B. JONES and W. J. THRON Continued Fractions: Analytic Theory and Applications, 1980 NATHANIEL F. G. MARTIN and JAMES W. ENGLAND Mathematical Theory of Entropy, 1981 GEORGE A. BAKER, Jr., and PETER R. GRAVES-MORRIS Pade Approximants Part I: Basic Theory, 1981 GEORGE A. BAKER, Jr., and PETER R. GRAVES-MORRIS Pade Approximants Part II: Extensions and Applications, 1981 ENRICO. G. BELTRAMETTI and GIANNI CASSINELLI The Logic of Quantum Mechanics, 1981 GORDON JAMES and ADALBERT KERBER The Representation Theory of the Symmetric Group, 1981 Other volumes in preparation

Analysis

Real Variables

Mathematics of Physics

Mathematics of Physics

Mathematics of Physics

Algebra

ENCYCLOPEDIA OF MATHEMATICS and Its Applications GIAN-CARLO ROTA, Editor Department of Mathematics Massachusetts Institute of Technology Cambridge, Massachusetts Editorial Board Shizuo Kakutani, Yale Janos D. Aczel, Waterloo Samuel Karlin, Stanford Richard Askey, Madison J. F. C. Kingman, Oxford Michael F. Atiyah, Oxford Donald E. Knuth, Stanford Donald Babbitt, U.C.L.A. Joshua Lederberg, Rockefeller Edwin F. Beckenbach, U.C.L.A. Andre Lichnerowicz, College de France Lipman Bers, Columbia M. J. Lighthill, Cambridge Garrett Birkhoff, Harvard Chia-Chiao Lin, M. I. T. Salomon Bochner, Rice Jacques-Louis Lions, Paris Raoul Bott, Harvard G. G. Lorentz, A ustin James K. Brooks, Gainesville Roger Lyndon, Ann Arbor Felix E. Browder, Chicago Marvin Marcus, Santa Barbara A. P. Calderon, Buenos A ires Peter A. Carruthers, Los Alamos N. Metropolis, Los Alamos S. Chandrasekhar, Chicago Jan Mycielski, Boulder S. S. Chern, Berkeley Steven A. Orszag, M. I. T. Hermann Chernoff, M. I. T. Alexander Ostrowski, Basle P. M. Cohn, Bedford College, London Roger Penrose, Oxford H. S. MacDonald Coxeter, Toronto Carlo Pucci, Florence Nelson Dunford, Sarasota,- Florida C. R. Rao, Indian Statistical Institute Fred S. Roberts, Rutgers F. J. Dyson, Inst. for Advanced Study Abdus Salam, Trieste Harold M. Edwards, Courant M. P. Schutzenberger, Paris Harvey Friedman, Ohio State Jacob T. Schwartz, Courant Giovanni Gallavotti, Rome Irving Segal, M. I. T. Andrew M. Gleason, Harvard Olga Taussky, Caltech James Glimm, Rockefeller Rene Thorn, Bures-sur-Yvette A. Gonzalez Dominguez, Buenos Aires John Todd, Caltech M. Gordon, Essex John W. Tukey, Princeton Peter Henrici, ETH, Zurich Stanislaw Ulam, Colorado Nathan Jacobson, Yale Veeravalli S. Varadarajan, U.C.L.A. Mark Kac, Rockefeller Antoni Zygmund, Chicago

GIAN-CARLO ROTA, Editor ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS Volume 16

Section: Algebra P. M. Cohn and Roger Lyndon, Section Editors

The Representation Theory of the Symmetric Group Gordon James

Adalbert Kerber

Sidney Sussex College Cambridge, Great Britain

University of Bayreuth Bayreuth, Federal Republic of Germany

Foreword by

P. M. Cohn University of London, Bedford College

Introduction by

G. de B. Robinson University of Toronto

CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521104128 © 1981 - Addison - Wesley Publishing Company, Inc. © Cambridge University Press 1985 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published by Addison Wesley 1981 First published by Cambridge University Press 1985 This digitally printed version 2009 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data James, G. D. (Gordon Douglas), 1945The representation theory of the symmetric group. (Encyclopedia of mathematics and its applications; v. 16. Section, Algebra) Bibliography: p. Includes index. 1. Symmetry groups. 2. Representations of groups. I. Kerber, Adalbert. II. Title. III. Series: Encyclopedia of mathematics and its applications; v. 16. IV. Series: Encyclopedia of mathematics and its applications. Section, Algebra. QA171.J34 512'.53 81-12681 ISBN 978-0-521-30236-4 hardback ISBN 978-0-521-10412-8 paperback

Contents Editor's Statement Section Editor's Foreword Introduction by G. de B. Robinson Preface List of Symbols

xi xiii xvii xxi xxiii

Chapter 1 Symmetric Groups and Their Young Subgroups 1.1 1.2 1.3 1.4 1.5

Symmetric and Alternating Groups The Conjugacy Glasses of Symmetric and Alternating Groups Young Subgroups of Sn and Their Double Cosets The Diagram Lattice Young Subgroups as Horizontal and Vertical Groups of Young Tableaux Exercises

Chapter 2 Ordinary Irreducible Representations and Characters of Symmetric and Alternating Groups 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

The Ordinary Irreducible Representations of Sn The Permutation Characters Induced by Young Subgroups The Ordinary Irreducible Characters as Z-linear Combinations of Permutation Characters A Recursion Formula for the Irreducible Characters Ordinary Irreducible Representations and Characters of An Sn is Characterized by its Character Table Cores and Quotients of Partitions Young's Rule and the Littlewood-Richardson Rule Inner Tensor Products Exercises

Chapter 3 Ordinary Irreducible Matrix Representations of Symmetric Groups 3.1 3.2 3.3

A Decomposition of the Group Algebra QSW into Minimal Left Ideals The Seminormal Basis of QSW The Representing Matrices vii

1 1 8 15 21 29 33 34 34 38 45 58 65 72 75 87 95 100 101 101 109 115

Contents

3.4 Chapter 4 4.1 4.2 4.3 4.4

The Orthogonal and the Natural Form of [a] Exercises

126 131

Representations of Wreath Products

132

Wreath Products The Conjugacy Classes of GwrSn Representations of Wreath Products over Algebraically Closed Fields Special Cases and Properties of Representations of Wreath Products Exercises

132 138

Chapter 5 Applications to Combinatorics and Representation Theory 5.1 5.2 5.3 5.4 5.5 Chapter 6 6.1

146 155 161

162

The Polya Theory of Enumeration Symmetrization of Representations Permutrization of Representations Plethysms of Representations Multiply Transitive Groups Exercises

163 184 202 218 227 237

Modular Representations

240

The/7-block Structure of the Ordinary Irreducibles of Sn and Aw; Generalized Decomposition Numbers The Dimensions of a/7-block; w-numbers; Defect Groups Techniques for Finding Decomposition Matrices Exercises

254 265 292

Representation Theory of Sn over an Arbitrary Field

294

7.1 7.2 7.3

Specht Modules The Standard Basis of the Specht Module On the Role of Hook Lengths Exercises

294 301 306 318

Chapter 8

Representations of General Linear Groups

319

Weyl Modules The Hyperalgebra Irreducible GL(m, F)-modules over F Further Connections between Specht and Weyl Modules Exercises

320 327 334

6.2 6.3 Chapter 7

8.1 8.2 8.3 8.4

240

341 346

Contents

ix

Appendix I: Tables LA I.B I.C I.D I.E I.F I.G I.H I.I

Character Tables Class Multiplication Coefficients Representing Matrices Decompositions of Symmetrizations and Permutrizations Decomposition Numbers Irreducible Brauer Characters Littlewood-Richardson Coefficients Character Tables of Wreath Products of Symmetric Groups Decompositions of Inner Tensor Products

348 348 356 368 380 413 430 436 442 451

Appendix II: Notes and References

459

II.A II.B II.C II.D Index

459 460 468 468 507

Books and Lecture Notes Comments on the Chapters Suggestions for Further Reading References

Editor's Statement A large body of mathematics consists of facts that can be presented and described much like any other natural phenomenon. These facts, at times explicitly brought out as theorems, at other times concealed within a proof, make up most of the applications of mathematics, and are the most likely to survive change of style and of interest. This ENCYCLOPEDIA will attempt to present the factual body of all mathematics. Clarity of exposition, accessibility to the non-specialist, and a thorough bibliography are required of each author. Volumes will appear in no particular order, but will be organized into Sections, each one comprising a recognizable branch of present-day mathematics. Numbers of volumes and sections will be reconsidered as times and needs change. It is hoped that this enterprise will make mathematics more widely used where it is needed, and more accessible in fields in which it can be applied but where it has not yet penetrated because of insufficient information. GIAN-CARLO ROTA

Foreword The theory of group representation has its roots in the character theory of abelian groups, which was formulated first for cyclic groups in the context of number theory (Gauss, Dirichlet, but already implicit in the work of Euler), and later generalized by Frobenius and Stickelberger to any finite abelian groups. For an abelian group all irreducible representations (over C) are of course 1-dimensional and hence are completely described by their characters. The representation theory of finite groups emerged around the turn of the century as the work of Frobenius, Schur, and Burnside. While it applied in principle to any finite group, the symmetric group Sn was a simple but important special case; —simple because its characters and irreducible representations could already be found in the rational field, important because every finite group could be embedded in some symmetric group. Moreover, the theory can be applied whenever we have a symmetric group action on a linear space. Perhaps the simplest example is the case of a bilinear form/(x, y). No theory is required to decompose/into a symmetric part: s(x, y)—f(x9 y)+f(y9 x) and an antisymmetric part: a(x, y)—f{x9 y) —f(y9x). These are of course just (bases for the 1-dimensional modules affording) the irreducible representations of S2, 1-dimensional because S2 is abelian. Taking next a trilinear form/(x, y9 z), we have again the symmetric and antisymmetric parts: s(x9y9z)=^f(ax9ay9az). a

a(x, y, z)= 2 sgnof(ox, oy, az), a

where a ranges over all permutations of x, y9 z. No other linear combination of / 's is only multiplied by a scalar factor by the S3-action (such a factor would have to be 1 or sgna, because every permutation is a product of transpositions), but we can find pairs of linear combinations spanning a 2-dimensional S3-module, e.g. p=f(x, y, z)+f(y, x, z)-f(z, y9 x)-f(z9 x9 y) q=f(z9 y9 x)+f{y9 z, x)-f(x9 y9 z)-f(x9 z, y) Here/? is obtained by 'symmetrizing' x, y and 'antisymmetrizing' x, z, and q

xiv

Foreword

is obtained by interchanging x, z in p. If (x, y) denotes the transposition of x, y etc., then we have (x,y)=p9 (x,z)q=p,

(x,z)p = q, (y,z)p=-p-q, (y,z)q = q.

(x,y)q=-p-q,

Thus we obtain the representation

< w ) - ( ° :[), {X.:,y)-(Z\ I). which is an irreducible 2-dimensional representation of S3. It was Alfred Young's achievement to find a natural classification of all the irreducible representations of Sn in terms of 'Young tableaux', which are essentially the different ways of fully symmetrizing and antisymmetrizing. The ^-symbols permuted are arranged in a diagram so that rows are symmetrized and columns antisymmetrized. In the above example we symmetrized x, y and antisymmetrized x, z; this is indicated by the tableau y

Young's derivation via tableaux was even more direct than Frobenius' and Schur's earlier method, using bialternants or S-functions, although these functions are useful in formulating combinations of representations such as plethysm. There have been many accounts of the theory, from various points of view, and often the original sources have been hard to follow. It is good to have a general treatment, —by two authors who have both made substantial original contributions, —which combines the best of previous accounts, and systematizes and adds much that is new. After a clear exposition of Young's approach (in modern terms) they present an improved version of Specht modules giving a characteristic-free treatment and leading to a practical algorithm for estimating dimensions. The applications to combinatorics include Polya's enumeration theory, and also the less well known work of Redfield, and there is a separate chapter on the connection with representations of the general linear groups. The comprehensive treatment, with helpful suggestions for further reading, very full references, various tables of characters, as well as the interesting historical introduction by G. de B. Robinson, will all help to make 'James-Kerber' the standard work on the subject. P. M. COHN

Cfl£ 3 f

Introduction In this introduction to the work of James and Kerber I should like to survey briefly the story of developments in the representation theory of the symmetric group. Detailed references will not be possible, but it seems worthwhile to glance at the background which has aroused so much interest in recent years. The idea of a group goes back a long way and is inherent in the study of the regular polyhedra by the Greeks. It was Galois who systematically developed the connection with algebraic equations, early in the nineteenth century. Not long after, the geometrical relationship between the lines on a general cubic surface and the bitangents of a plane quartic curve aroused the interest of Hesse and Cayley, with a significant contribution by Schlafli in 1858 [1, Chapter IX].* Jordan in his Traite des Substitutions, 1870 [2], and Klein in his Vorlesungen uber das Ikosaeder, 1884 [3], added new dimensions to Galois's work. The first edition of Burnside's Theory of Groups of Finite Order appeared in 1897, just at the time when Frobenius's papers in the Berliner Sitzungsberichte were changing the whole algebraic approach. With the appearance of Schur's Thesis [4] in 1901, the need for a revision of Burnside's work became apparent. Burnside began his preface to the second edition, which appeared in 1911 [5], with the comment: "Very considerable advance in the theory of groups of finite order has been made since the appearance of the first edition of this book. In particular the theory of groups of linear substitutions has been the subject of numerous and important investigations by several writers...." His preface concludes with the remark: "I owe my best thanks to the Rev. Alfred Young, M.A., Rector of Birdbrook, Essex, and former Fellow of Clare College, Cambridge, who read the whole of the book as it passed through the Press. His careful criticism has saved me from many errors and his suggestions have been of great help to me." Alfred Young was born in 1873 and graduated from Cambridge in 1895. His first paper, "The irreducible concomitants of any number of binary quartics," appeared in the Proceedings of the London Mathematical Society in 1899. It had been refereed by Burnside, who told him to read the works of Frobenius and Schur; unfortunately Young knew no German, so it was not till after the war that he was able to incorporate their ideas in his important QSA series. * References will be found at the end of this Introduction.

xviii

Introduction

My own contact with Alfred Young began in 1929. I graduated from the University of Toronto in 1927 and was much interested in geometry, owing largely to the presence on our staff of Jacques Chapelon from Paris. I was fortunate in obtaining a small scholarship at St. John's College, Cambridge, where my first supervisor was M H. A. Newman. Under his guidance I began to read topology. No group theory was taught in Toronto or Cambridge in those days, but its significance in topology fascinated me. Soon this became apparent to Newman, and he arranged for me to be transferred to Alfred Young as a graduate student. Young came in to Cambridge once a week to lecture. He and his wife stayed at the Blue Boar Hotel, just across the street from St. John's, where I would go to visit him. The geometrical aspects of group theory continued to interest me, and I attended Baker's tea party every week. This was where I met Donald Coxeter and several other geometers to whom I refer in the Introduction to Young's Collected Papers, published in Toronto [6], 1977. After earning my Ph.D. in Cambridge in 1931, I returned to the University of Toronto. My work on the symmetric group continued, and with the cooperation of J. S. Frame and Philip Hall, yielded the dimension formulae for the irreducible representations of Sn and GL^ over the real field. Richard Brauer was on staff in Toronto 1935-1948, and his interest in representation theory was responsible for much of the development which took place in those years and later, while he was at Ann Arbor 1949-1952, and Harvard 1952-1978. In 1958 I was invited to lecture at the Australian universities, and my Representation Theory of the Symmetric Group appeared in 1961. In 1968 my wife and I went to Christchurch, New Zealand, for three months, and it was during this period that I became interested in the application of group theory to physics. W. T. Sharp in Toronto had obtained his Ph.D. with Wigner in Princeton, and I made contact with Wybourne in New Zealand and with Biedenharn in the U.S., and attended a seminar in Bochum in West Germany in 1969. It was there that I met Adalbert Kerber and many other interesting people. Not long after, I was in touch with Gordon James, who got his Ph.D. in Cambridge with J. G. Thompson. Then when the representation theory gathering was held in Oberwolfach in 1975 I had a chance to talk with many group theorists whose writings I had read, but had never met. Afterwards my wife and I paid a brief visit to the Kerbers in historic Aachen. It was in the autumn of 1975 that Gordon James came to spend a year at the University of Toronto. He and Kerber had begun to work on this book and we had many conversations; Kerber was largely interested in wreath products, while James had begun writing his considerable number of papers on modular theory. A number of errors had appeared in the decomposition matrices at the end of my book [7], and James has done much to improve their construction in this volume.

References

xix

In April of 1976 Foata organized another gathering of group theorists in Strasbourg. He did a beautiful job, exploiting the charm of the city and its university to bring together a large number of speakers [8] on various aspects and applications of the symmetric group. Having been invited by Professor McConnell, I gave a repeat performance of my Strasbourg talk in Dublin. This was my first visit to Ireland, and it gave me much pleasure to see J. L. Synge, who had been on our staff in Toronto for many years. It was in June 1978 that T. V. Narayana of the University of Alberta in Edmonton arranged a gathering at the University of Waterloo. He had become involved in Young's work, and his volume Lattice Path Combinatorics with Statistical Applications was published in our Exposition Series in 1979. The proceedings of Young Day has just appeared [9] with an introduction by J. S. Frame and a paper generalizing the hook-formulae for OB(2). The last gathering in Oberwolfach which I attended was in January 1979. This book was well on its way but we all regretted that publication would be so long delayed. Its content contributes much to complete the picture, from the point of view of the representation theory of Sw, but there remains the question raised by Frame's work:—Could there be a degree formula for the irreducible representations of Sn or GLd over a finite field? Future research may provide the answer. In conclusion, let me refer to The Theory of Partitions [10] by Andrews, which has appeared in this series. Professor Rota's comment is worth quoting: "Professor Andrews has written the first thorough survey of this many-sided field. The specialist will consult it for the more recondite results, the student will be challenged by many deceptively simple facts, and the applied scientist may locate in it a missing identity to organize his data." When Young's Collected Papers came out, Andrews wrote a most interesting review of his work [11], listing 121 papers based on the original ideas of this remarkable man. The accompanying portrait of Alfred Young was sent to me by Professor Garnir. It has given me much pleasure to work with the authors of this book, and I wish its readers every satisfaction, as well as the best of luck in further developing these ideas.

References 1. G. Miller, H. Blichfeldt, and L. Dickson, Finite Groups, Dover Publications, New York, 1961. 2. C. Jordan, Traite des Substitutions, Paris, 1870. 3. F. Klein, Vorlesungen tiber des Jkosaeder, B. G. Teubner, Leipzig, 1884. 4. I. Schur, Inaugural—Dissertation, Berlin, 1901. 5. W. Burnside, Theory of Groups of Finite Order, Cambridge University Press, Cambridge, 1911.

Introduction

6. A. Young, Collected Papers, University of Toronto, 1977. 7. G. de B. Robinson, Representation Theory of the Symmetric Group, University of Toronto, 1961. 8. D. Foata, Combinatoire et Representation du Group Symetrique, Strasbourg 1976, Springer-Verlag (579), 1977. 9. T. V. Narayana, Young-Day Proceedings—Waterloo 1978, Marcel Dekker (57), 1980. 10. G. E. Andrews, The Theory of Partitions, Addison Wesley, Advanced Book Program, 1976. 11. G. E. Andrews, Bull. Am. Math. Soc, 1, 989-997 (1979). G. DE B. ROBINSON

Preface The purpose of this book is to provide an account of both the ordinary and the modular representation theory of the symmetric groups. The range of applications of this theory is vast, varying from theoretical physics through combinatorics to the study of polynomial identity algebras, and new uses are still being found. So diverse are the questions which arise that we feel justified in hoping the reader might find that some part of our text inspires him to undertake research of his own into one of the many unsolved problems in this elegant branch of mathematics. There are several different ways of approaching symmetric group representations, and while we have tried to illuminate parts of the theory by giving more than one description of it, we have made no effort to cover every view of the subject. The ordinary representation theory of the symmetric groups was first developed by Frobenius, but the greatest contribution to the early material came from Alfred Young. Since Young's main interest lay in quantitative substitutional analysis, it is difficult for a modern mathematican to understand his papers. The reader is referred to the book Substitutional Analysis by D. E. Rutherford for a pleasant account of a great part of Young's work. Both Frobenius's and Young's collected works are now available. We include an account of the group algebra and its idempotents, along the lines pursued by Young, since the symmetric group is one of the very rare cases where many aspects of general representation theory can be described explicitly. This also helps us to motivate the introduction of many combinatorial structures which turn out to be useful. The combinatorics are themselves a fascinating and fruitful basis for further study, and they continue to inspire many research papers. The development of the modular representation theory of symmetric groups was started by T. Nakayama, who derived some /7-modular properties of symmetric groups Sn, for n L« Lf m+ n m-(-n\in mn Mn M^ n na nx n) n[k] n(/c) nw W ^o p(k;m,n) p(n) p'{ n) p*(n) P(n) pW9 pik)^

The number of nodes in the hook //?. A certain "hook length" 2 p e / f «p The horizontal group of ta The horizontal group olta The(/,y)-hookof[a] Intertwining number The identity mapping of {1,2,...,«} The simple two sided ideal of QSW corresponding to the partition a The identity representation of Sx The leg length of the hook H? ®n^oL(n) pr (1) ® • • - ® W(1) (w times) cyaL(W> The left ideal of Q Sn generated by ef {1,2,..., m + n) {m+\,...9m-\-n} {subsets of n The set of all A>tuples over n The set of all injective fc-tuples over n {1,2,3,...} {0,1,2,...} The number of proper partitions of k into —w-£2,

having the property that (-^n}.

So consists of one element only (as does S,). An easy induction shows that the following is true: 1.1.6

Vn>0

(\Sn\=n\).

A permutation fl"ESn is written down in full detail by putting the images

1.1 Symmetric and Alternating Groups

7r(/) in a row under the points /En, say

7r=

(^(i)

•••

This will sometimes be abbreviated by

*(,-) Hence, for example, s

_ ( f/ / l

b3

22

~Ul

\ f1l 33 W

22

33 W \/ 1 2

2 3 A 2 1 3J'l.3

1 1

31 ( 1 2/ ' \ 2

2 3

2 3

2

3\(1 2 W'U 1

3 l/'

3\] 2)}'

The points \,...,n which form the first row need not be written in their natural order; e.g. (1 \2

2 3

3\ 1/

(2 \3

1 2

3\ 1/'

With this in mind, we call a permutation of the form 1.1.7

h ' ' "'#•-1 l

hh'" r

l

l

l

r

\

l

r+\

r+\

cyclic or a cycle. In order to emphasize r, we also call 1.1.7 an r-cycle. A shorter notation for the cycle 1.1.7 is 1.1.8

( / „ . . . , i r )(/ r + 1 ) •••(;„),

where the points which are cyclically permuted are put together in round brackets. For example

\

|)-(2.3KD.

Commas which separate the points may be omitted if no confusion can arise (e.g. if «(6

4)(8)

-

The set of disjoint cycles is uniquely determined by the given permutation.

1.1 Symmetric and Alternating Groups

5

For a general TTESW let C(IT) be the number of disjoint cyclic factors including 1-cycles, let lv (1 ^V^C(7T)) be their lengths, and choose for each v an element^ of the v-ih cyclic factor. Then

v=\

This notation becomes uniquely determined if we choose the jv so that the following holds: x1

0)

(),

(ii)VKiKc(ir)

U(jv))> U l c and TTh^sgn IT.

1.2

The Conjugacy Classes of Symmetric and Alternating Groups

We shall describe the conjugacy classes of Sn. In order to do this, we first of all note how p7Tp~l is obtained from IT. Since

we get pTTp~x from IT by an application of p to the points in the cyclic factors of the same m. For if

then by 1.2.1 we have

We notice that under this process of applying p to the points, the brackets remain invariant, so that the cyclic factors of p7rp~l (in cycle notation) are of the same lengths as those of 77.

1.2 Conjugacy Classes of Symmetric and Alternating Groups

of

9

On the other hand, let m and a be permutations which are both products C(TT) cyclic factors of the same lengths lv, 1^V^C(7T), say

v=\

while

v~ 1

We now put

Then by 1.2.1 we obtain

This shows that two permutations are conjugate if and only if they have the same cycle structure. In order to make this more precise, we introduce the notion of a partition of n. A sequence of nonnegative integers

is called a {proper) partition of n if and only if it satisfies

12 2

°°

(ii) 2 «/=«. The a, are called the parts of a. The fact that a is a partition of n is abbreviated by a\-n.

If avn, then by 1.2.2 (ii), there is an h such that at = 0 for all i>h. We may take the liberty of shortening a as follows:

10

Symmetric Groups and Their Young Subgroups 1.2

(normally, we choose h such that ah>0, ah+]=0). We list below the partitions of the first few nonnegative integers, using this convention: «=0

(0)

n=l

(1)

»=2 «=3 n=4

(2),(1,1) (3),(2,1),(1,1,1) (4),(3,1),(2,2),(2,1,1),(1,1,1,1).

The number p(n) of partitions of n grows rapidly with n. E.g.

= 627, /?(50) = 204226, ^(100)= 190569292. A table for p(n) («^100) can be found in the book of G. E. Andrews [1976]. The following notation is useful in the case when several nonzero parts of a are equal, say at parts are equal to /,

If at = 0 , then ia> is usually left out. For example (3,2,1 2 ) = (3,2,1,1,0,0,...). If now 77 is an element of Sw, then the ordered lengths a, (77), of the cyclic factors of TT in cycle notation form a uniquely determined partition of n, which we call the cycle partition of 7r, and which we denote by 1.2.3

a(w): = (a 1 (w),...,a c(ir) (7r)).

The corresponding w-tuple consisting of the multiplicities of parts of a(7r), i.e. 1.2.4

a{TT): =

{ax{m),...,an{v))

is called the cycle type of IT. Correspondingly we call a: = (a,,..., an) a type of n if and only if (i)

Vl5,

(5,1)1-6, (7,3)1-10, (9,1)1-10, , (9,5>14.

In each of them the number of components congruent to 3 modulo 4 is 0 or 2, so this number is even. Hence the standard conjugator is even in each case, by (i). This shows that the groups mentioned in the statement are in fact ambivalent. (iii) If the standard conjugator £ of TT is odd, then the An-class CA(IT) in question is not ambivalent. For if aEA n satisfies OTTO~1 = TT~1, then

which contradicts CS(TT) —CA{IT), since a~ 1 |ES n \A /i . (iv) It remains to show that for each «£{0,1,2,5,6,10,14} there are partitions a with pairwise different and odd parts a^O such that the number of at satisfying a • = 3 (4) is odd. (a) (b) (c) (d)

n = 4k, A; EN: (4k—1,1) has the desired properties. n = 4k+\, 2

a^p.

It will be useful to have a characterization of partitions a and P of n which

Symmetric Groups and Their Young Subgroups 1.4

24

are neighbors with respect to ^ , a situation which we denote by a are Young subgroups of Sn which correspond to a and /?', and if the groundfield F has charF^2, then i(ISa^Sn, ASp, TSn)^O */ and only if

28

Symmetric Groups and Their Young Subgroups 1.4

The proof of this theorem given by Ruch and Schonhofer does not use the Gale-Ryser theorem but another characterization of a^P in terms of so-called Young tableaux. A Young-tableau ta with Young diagram [a] (sometimes called an a-tableau for short) arises from [a] by replacing the nodes X of [a] by the points / of n = { l , . . . , n). For example, here are two of the 7! Young tableaux with diagram [3 2 ,1]: 1 4 2 5 3

6 7

5 2 7

4

1 6 3

The tableau obtained by replacing the nodes by 1,2,... in order down successive columns is denoted by *{*: 1 2 1.4.19

a\ + 1

t?: =

a[+a2 1.4.20 LEMMA, / / a , fivn and ta is an a-tableau, then a^fi if and only if there exists a ^-tableau t& such that any two points which occur in ta in the same row occur in t& in different columns. Proof Assume the existence of ta, t^ with the given property. Then 6. We would like to derive a result, which in a sense reverses 2.2.17, and which will turn out to allow a representation theoretical proof of the Ruch-Schonhofer theorem. We have already seen that i(ISa tS w , ISfi TSJ is equal to the number of matrices over N o with a as vector of row sums and /? as vector of column sums. We can apply this if we happen to know the number of such matrices. The following lemma gives the number of matrices in a particular case: 2.2.18 LEMMA. / / rl9 r2, cl9 and c2 are nonnegative integers with the property rx-\-r2—cxJrc2, then the number of 2X2 matrices over No with row sums rl9 r2 and column sums cv c2 is equal to 1 +min{r 1 , r2, c l5 c2}.

2.2 The Permutation Characters Induced by Young Subgroups

43

Proof. If for example rx —mm{rx, r2,cx,c2}, then we have the following 1 +rx choices for the entries of the first row of such a 2X2 matrix:

Each of these choices yields exactly one 2X2 matrix with row sums rt and column sums ci9 since rx was assumed to be the minimum, so that a suitable second row can be found and is uniquely determined. In the case when r2, cl9 or c2 is the minimum, an analogous argument yields the statement. • This helps in the proof of 2.2.19

LEMMA.

/ / a — {ax,a2)\-n, a2 > 0 , and a*:=(ax + 1, a2 — l), then

(ii) the dimension fa of [a] satisfies

«2

Proof (i): 2.2.18 yields for the inner product of the generalized character -£«* with itself:

Hence | a — ^a* is ± 1 times an irreducible character of Sn. Since (/S t t TS^,[a])=l, but (7Sa*TSM,[a]) = 0 (by 2.1.10), we obtain £«-£«*=£", as stated, (ii): The statement follows from the dimensions

by an application of (i). (iii): This part follows immediately from part (i), by induction on h; then for all T T E S N \ S A , A —id+ 77 has a negative part. Therefore, the sum in 2.3.8 is finite and

xX= 2

48

Ordinary Irreducible Representations and Characters 2.3

Thus xX is a generalized character of Sn. In particular, if A: = a\-n, then xa is clearly the character of the determinant \[at +j—i]\, so that it is our aim to

prove

a

X

=r.

We divide the proof of this basic result into several steps. A preliminary lemma gives an important property of xA2.3.9 LEMMA. / / A is a composition of n, and

Proof. We put T: = (/, / + 1 ) E 5 ^ . Then we have if ), i — id + 77)(/) This implies |M-id+*r = |A-id+,r j

2.3.10

LEMMA.

j=£i9i+l,

if j=i, if j—i-\-\.

so that

Suppose that A is a composition of' n = m + k. Then

Proof (i): Both sides of the equation equal zero unless A is an improper partition of m + k. Assume, therefore, that At=m + &. (The sum over \x has only finitely many nonzero terms, since $ x ~ /4 #^ /A =0 unless A —jui=m.) Mackey's subgroup theorem then yields

By 1.3.10 the double-coset Sm XSkirSx is characterized by the 2X« matrix

It is therefore uniquely determined by

2.3 Irreducible Characters as Combinations of Permutation Characters

Furthermore, as (Sm XSk)H7TSXTT~l

49

= Sx_^ XS^ we have

The character of this representation is

and so (i) is proved. (ii): The definition 2.3.8 of xX yields

and by (i), this equals

2 If A is a composition of n, and \i is a composition of /c, we define A/

2.3.11

2

x i d i d

We claim that the following is true: 2.3.12

LEMMA.

/ / X is a composition of m + k, then

Proof. The proof of 2.3.10(ii) has already shown that

Replacing ju by JUOTT we get (as £/XO7r =£**) x X

lS m X S , = 2 sgn77 2 ^ - i d

Now, for each ]ix, /A, is eventually zero, and thus there exist uniquely determined /?N k and a G 5 ^ such that (/? — id)oa = ]Lt—id and

50

Ordinary Irreducible Representations and Characters 2.3

This enables us to proceed as follows (recall that all the sums are finite, so there is no problem about rearranging): XXlSmXS^=

2

sgnir

77G5M

2 O65N

ft

sgna^- i d + »~

frk The last equation holds because x* = 0 if A- - * = j8 /+ , - (/ + 1) (by 2.3.9).

•

The lemma shows the importance of the generalized characters xX/f* °f 2.3.13

THEOREM.

Ifat-n, k^n, and fivk, then

(i) xa/li ^ 0 only if at >/?, for all i,

(ii) (

X

, € )

)= ff ! '/a:i

I 0 othe otherwise.

Proof, (i): We consider the determinant by which x a / / s is defined (cf. 2.3.11). It is (cf. 2.3.4)

As the sequences a —id and >8 — id are strictly decreasing, an entry [ccl:— i — (j8y—y)] = 0, i.e. ai—i — (Pj—j)fii >a2 >P2 ^a3 ^ * * *, then this determinant has l's along its main diagonal and O's below, and hence it is equal to 1. Otherwise it is not of this triangular form, so that by the monotonicity of the sequences a —id and /? — id two columns are equal, and hence the determinant must vanish in this case. • This result turns out to be crucial in the proof of 2.3.14 YOUNG'S RULE (FIRST VERSION). For each AN n such that Xt —0 when i>n and every partition a of n, we have that the inner product (x a , £A) is equal to the number of (n— \)-tuples (/? (1) ,..., /?(/1~1}) such that (ii) VI' (iii) V/> 1, i> 1 (j8/° j8{"-1>>«2>)8

1=1

Another application of 2.3.12 and 2.3.13 yields

where the sums have to be taken over fHn~V

and

jg(»-2) subject to the

52

Ordinary Irreducible Representations and Characters 2.3

following conditions: 2 (i)j8$""•>

| >

Further iterations yield

(x",€x)=

where the sum is taken over all (n— l)-tuples (/J (1) ,..., /?(w~1}) subject to the conditions described in the statement. • We are now in a position to prove the main result of this section: 2.3.15

THEOREM.

For each a\-n

we have that

xa=r, the character of the ordinary irreducible representation [a] of Sn. Thus, in particular, the set

is the complete set of ordinary irreducible characters of Sn. Each fa can be expressed in the determinantal form

as a linear combination of permutation characters £ \ \\=n, with coefficients 0, ± 1 . We can therefore express the representation [a] itself as a generalized "representation" in the following determinantal form:

subject to the conventions [r]: — 1 if r—0, and [r]: = 0 if rn), (xa> implies a^X: This follows from Young's rule, for condition 2.3.14(iii) yields $D= i 8 3 Xx (by P^=0); £ (2) b\,+A together with ax^pf\ a2>fe2) yields , 2 2 2 x 2) 2) { $ +A2 (by $ 2 ) =0), and so on. (ii) (x°, £ a ) = l : By Young's rule ( x M a ) is equal to the number of (w-l)-tuples (/?(1),...,/?("~1)) which satisfy (see (i))

Thus ((«!),(«!,a 2 ),...,(«!,...,«„_!)) is the only possible (n— l)-tuple. (iii) ( x a , x a ) = l : If( x a ,r~ i d + 7 r )^O,thena-id-h77t=«anda^a by (i). But a —id+ 7r^ a, as it is very easy to see, and so a — a — id + 77 and 77= 1. Together with part (ii), this shows that £a is the unique summand £A of X a for which (x*, £ x )^0. Therefore, ( X a , Xa) = (x a , € a )= 1. (iv) In order finally to identify x a with fa, we consider the determinantal form \[at — /H-y"]| which shows that [a] is contained (use 2.2.22) in the main diagonal term only. Thus (x a , ?")—1. But (iii) shows that x a is ± a n irreducible character, and so x a = fa• A numerical example is

[2] [3] 1 [1] But

[2][1]=/5 2:

2.3.16

(ii) f •) /5> a partition of n, then we have for the restriction of [a] to the stabilizer Sn_x of the point n

i=

2

[«'"].

>a / + 1

On the other hand, if Sn denotes the stabilizer of the point n+\ in Sn+], then we have for the induced representation

[«]tsn+1=

2

[«'+]

For example

while

This branching theorem shows how to evaluate f a(ir) if m contains 1-cycles and if the character table of Sn_l is known. In the case where m does not contain 1-cycles, we know £a(7r) only if IT is an w-cycle (cf. 2.3.17). Hence if 77 contains a A>cycle, say, then a formula would be welcome which expresses fa(7r) in terms of certain f^(p), f$\-n — k, p arising from IT by cancelling this A:-cycle. We shall derive such a formula, and it should be clear from the foregoing that we need first to restrict from Sn down to Sn_A:XSA:, the subgroup of elements which leave the subsets n —k and n\n —k invariant. An immediate corollary of 2.3.10(ii) is 2.4.4 LEMMA. Suppose that X is a composition of n = m + k. If'TTESW contains a k-cycle, while p E Sm has the cycle type

then

60

Ordinary Irreducible Representations and Characters 2.4

In order to apply this we remember that ^k) T^O if and only if \i has the form in which case £fc — 1. Therefore, 2.4.5

Y X U ) = 5 y (X

Now we consider the summands in 2.4.5 for the case when X — a is a partition of n. An m-fold application of 2.3.9 yields 2.4.6 ( a , , . . . , a , _ , , a , - £ , « , + , , . . . ) _ / _ j \m

( a , , . . . , a , _ j , a I + 1 - 1,..., a/

+w

- 1, a ; - k + m , a l

+ m +

,,...)

It is not difficult to see that the existence of an mX) such that the composition on the right-hand side of 2.4.6 is a proper partition of n — k is equivalent to the fact that the corresponding diagram is obtained from [a] by cancelling the rim part R"j of a (uniquely determined) hook ///* of length k of [a]. In this case m = /I", and we obtain

If no such m exists, it follows from the considerations above and 2.3.8 that the character in 2.4.6 vanishes. This yields the desired recursion formula for

2.4.7 THE MURNAGHAN-NAKAYAMA FORMULA. / / a is a partition of n, if i G n , and TTES^ contains a k-cycle, while pESn_k is of cycle type (al(n)9...,ak_l(v),ak(v)-l,ak+i(w),...,an_k(ir)), then we have the following recursion formula for f "(TT):

£"(*)= 2 (recall that f[0] = 1). For example

+ £((1234)(56))

2.4 A Recursion Formula for the Irreducible Characters

61

It is clear that a diagram [a] has exactly one hook of length h"x. Thus yet

— f— i ^ i ^ M a M f i

and a repeated application of this argument yields 2.4.8 COROLLARY.

^hau_h%k)={-\)^h"u and that this implies 2.4.9 COROLLARY.

tf^0=>p^(h«x,...,hakk).

This shows once again the importance of the diagram lattice. Later on we shall need a few results on lowest dimensions of ordinary irreducible representations of symmetric groups. There are many results known, we would now like to derive a few of them by various applications of the branching theorem. 2.4.10

THEOREM.

(i) For each n, [n] and [\n] are the only one-dimensional ordinary representations of Sn. (ii) For 2 < r c ^ 4 , the lowest dimension =£\ of an ordinary irreducible representation ofSn is n—\ (whilefor n:—4,[22] has dimension 2 — n — T). For 2 ^ « ^ 6 , [n— 1,1] and [2, \n~2] are the only ordinary irreducible representations ofSn which are of dimension n—\ (while for n: = 6,[32] and [23] are other irreducible representations of dimension 5 — n— 1). (hi) For 2^n^5 there is no ordinary irreducible representation [a] of Sn which is of a dimension fa such that

(while for n: = 5 we have [3,2] and [22,1], which are of dimension 5 — n, and [3,1 2 ] of dimension 6 = n H-1). Proof (i): This part is an easy consequence of 1.1.25 and 1.1.26. If we prefer to avoid this argument, we may proceed by induction on n as follows. If [a], avn, is one-dimensional, the same holds for [a]lS n _j. Hence, by the induction hypothesis, [ a U S , , ^ coincides with [n-\] or [I 11 " 1 ], and the branching theorem yields

But 2.3.21 has shown that both [ H - 1 , 1 ] and [2, T~ 2 ] are of degree which finishes the proof of part (i).

H-1,

62

Ordinary Irreducible Representations and Characters 2.4

(ii): An inspection of the character table shows that the statement holds for «^8. We can therefore assume a\-n>9 and (a) If [a]lSAI_1 contains one-dimensional constituents, we obtain from (i) and the branching theorem that

and 2.3.21 shows that both [n — 1,1] and [2, \n~2] are of dimension n—\. (b) If [ a H S ^ ! does not contain any one-dimensional constituent, then by induction it must be irreducible, for Sn_1 has no irreducible representation [/?] such that 1 6. Hence the induction hypothesis yields the contradiction

(iii) follows immediately from the proof of (ii).

•

The proof of this theorem shows clearly that analogously one can derive further results concerning ordinary irreducible representations of low dimension. We next undertake to extend the Murnaghan-Nakayama formula so that it can be applied to the generalized characters x x//1 introduced in 2.3.11. To do this we need the following result: 2.4.11 LEMMA. / / X is a composition of n + r, v a composition of r, and — n, then (ii) and

if TT"ESW contains

a k-cycle,

/Lti= k

while

p£=S m is of cycle

type

2.4 A Recursion Formula for the Irreducible Characters

63

Proof, (i): We obtain from 2.3.11 that

so that an application of 2.3.10 yields

the last equation coming from 2.3.11. (ii): As

we obtain from (i)

= 2 x(X"")A(p)€f*)Again we remember that £^=^0 if and only if /x is of the form JU = (0,...,0, A:,0,...), in which case £ ^ = 1. Therefore, the following must be true for X, j ' , TT, and p as in 2.4.11: XX/v(v)= 2

2.4.12

x ^'-'

A

'-'^-^ A -—)A(p).

1=1

In order to derive from this the desired generalization of the MurnaghanNakayama formula 2.4.7, we associate with xa//*> ai-m +fc,i8i-fc, Pi - » S P ( / i ) : \, then the values of £a± are

for a suitable numbering of the constituents of [a] IA n, while on all the other classes with cycle partitions y^h(a) we have

and f" is an even integer. Proof. We consider the following generalized character of An:

68

Ordinary Irreducible Representations and Characters 2.5

where

i.e., fa± denotes the character contragredient to fa±. We would like to show that 4. The character tableZ rt of G yields the coefficients a Zpy> fr°m which we can see that C ( 2 ' r } generates G. Hence by (iii) for n>4 the image of [n— 1,1] is a faithful representation of G as an irreducible group of reflections. These groups are known (see e.g. Benson and Grove, [1971, Theorem 5.3.1]), so that we obtain from |G| = |SJ = «! that V«>4

(G-SJ.

(v) G—Snifn