06364 - Representation Theory of the Symmetric Group [James-Kerber].pdf

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GIAN-CARLO ROTA,
Edilor
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
GordonJames
Sidney Sussex College
Cambridge, Great Britain
AdalbertKerber
University of Bayreuth
Bayreuth, Federal Republic of Germany
Foreword by
P. M. Cohn
University of London, Bedford Col1ege
Introduction by
G. de
B.
Robinson
University of Toronto
••
....
1981
Addison-Wesley Publishing Company
Advanced Book Program
Reading, Massachusetts
London' Amsterdam- Don Mills. Ontario· Svdnev- Tokyo
Library of
Congress
Cataloging in Publication Data
James. G. D. (Gordon Douglas). 1945-
The representation theory of the symmetric group.
(Encyclopedia of mathematics and its applications;
v. 16. Section. Algebra)
Bibliography: p.
Includes index.
I. 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 0-201-13515-9
AACR2
American Mathematical Society (MOS) Subject Classification Scheme (1980): 20C30
Copyrightc 1981 by Addison-Wesley Publishing Company. Inc.
Published simultaneously in Canada.
All rights reserved. No part of this publication may be reproduced. stored in a retrieval system.
or transmitted. in any form or by any means. electronic. mechanical. photocopving. recording.
or otherwise. without the prior written permission of the publisher. Addison- Wesley Publishing
Company. Inc .. Advanced Book Program. Reading. Massachusetts 01867. U.S.A.
Manufactured in the United States of America
Contents
Editor's Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Section Editor's Foreword
Introduction by G. de
B.
Robinson
Preface
List of Symbols
XIII
xv
xvii
xxi
xxiii
Chapter 1 Symmetric Groups and Their Young Subgroups . . . . . . . . . . . 1
1.1 Symmetric and Alternating Groups
1.2
The Conjugacy Classes of Symmetric and Alternating
Groups
1.3 Young Subgroups of S, and Their Double Cosets
1.4
The Diagram Lattice
1.5 Young Subgroups as Horizontal and Vertical Groups of
Young Tableaux
Exercises
1
8
15
21
29
33
Chapter 2 Ordinary Irreducible Representations and Characters of
Symmetric and Alternating Groups . . . . . . . . . . . . . . . . . . .34
2.1
The Ordinary Irreducible Representations of
S,
2.2
The Permutation Characters Induced by Young
Subgroups
2.3 The Ordinary Irreducible Characters as Z-linear
Combinations of Permutation Characters
2.4
A Recursion Formula for the Irreducible Characters
2.5 Ordinary Irreducible Representations and Characters
of
An
2.6
S, is Characterized by its Character Table
2.7 Cores and Quotients of Partitions
2.8 Young's Rule and the Littlewood-Richardson Rule
2.9 Inner Tensor Products
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 3 Ordinary Irreducible Matrix Representations of
Symmetric Groups
3.1 A Decomposition of the Group Algebra
QS
n
into
Minimal Left Ideals
3.2 The Seminormal Basis of
QS
n
..•••••••............
3.3 The Representing Matrices
va
34
38
45
58
65
72
75
87
95
100
101
101
109
115
viii
Contents
3.4
The Orthogonal and the Natural Form of
(a
1
Exercises
Representations of Wreath Products
Wreath Products
The Conjugacy Classes of GwrS
Representations of Wreath Products over Algebraically
Closed Fields
Special Cases and Properties of Representations of
Wreath Products
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ii
•••••••••••••••••••
126
131
132
132
138
146
155
161
Chapter 4
4.1
4.2
4.3
4.4
Chapter 5 Applications to Combinatorics and Representation
Theory
5.1
5.2
5.3
5.4
5.5
The P61ya Theory of Enumeration
Symmetrization of Representations
Permutrization of Representations
Plethysms of Representations
Multiply Transitive Groups
Exercises
162
163
184
202
218
227
237
Chapter 6
6.1
6.2
6.3
Modular Representations . . . . . . . . . . . . . . . . . . . . . . ... 240
The p-block Structure of the Ordinary Irreducibles of
SIIand All; Generalized Decomposition Numbers
The Dimensions of a p-block; u-numbers; Defect
Groups
Techniques for Finding Decomposition Matrices
Exercises
240
254
265
292
Chapter 7
7.1
7.2
7.3
Representation Theory of
Sn
over an Arbitrary Field . . . . . .294
Specht Modules
The Standard Basis of the Specht Module
On the Role of Hook Lengths
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Representations of General Linear Groups
Weyl Modules
The Hyperalgebra
Irreducible GL(m.
F)-modules
over
F
Further Connections between Specht and Weyl
Modules
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
294
301
306
318
319
320
327
334
341
346
Chapter 8
8.1
8.2
8.3
8.4

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