**Graduate Studies in Mathematics**

Volume: 10;
1996;
266 pp;
Hardcover

MSC: Primary 20; 22;

Print ISBN: 978-0-8218-0453-7

Product Code: GSM/10

List Price: $45.00

AMS Member Price: $36.00

MAA Member Price: $40.50

**Electronic ISBN: 978-1-4704-1142-8
Product Code: GSM/10.E**

List Price: $45.00

AMS Member Price: $36.00

MAA Member Price: $40.50

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# Representations of Finite and Compact Groups

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*Barry Simon*

Barry Simon, I.B.M. Professor of Mathematics and
Theoretical Physics at the California Institute of Technology, is the
author of several books, including such classics as

The exposition in the book centers around the study of representation
of certain concrete classes of groups, including permutation groups and
compact semisimple Lie groups. It culminates in the complete proof of
the Weyl character formula for representations of compact Lie groups and
the Frobenius formula for characters of permutation groups. Extremely
well tailored both for a one-year course in representation theory and
for independent study, this book is an excellent introduction to the
subject which, according to the author, is unique in having “so
much innate beauty so close to the surface”.

#### Readership

Research mathematicians and graduate students.

#### Reviews & Endorsements

Contains a very good explanation of representation theory of finite and compact groups and can be recommended to everyone for learning or teaching representation theory.

-- Zentralblatt MATH

This is indeed a nice book … I would recommend it precisely for the graduate course [that] I am teaching now, “Representation Theory” … I very much like the hands-on approach and the very explicit formulae that are given … Professor Simon has done an excellent job on this beautiful material.

-- Tudor Ratiu, University of California, Santa Cruz

Can be recommended as a base for courses about representations of finite groups and finite-dimensional representations of Lie groups.

-- Mathematical Reviews

This book closes a gap that impeded instruction in the principles of the theory of group representations in the past … it would also tend to close the gap between the interested student of the subject and the advanced researcher keen on bare facts … a most readable account of a subject of enduring fascination.

-- Monatshefte für Mathematik

#### Table of Contents

# Table of Contents

## Representations of Finite and Compact Groups

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Introduction xi12 free
- CHAPTER I. Groups and Counting Principles 114 free
- CHAPTER II. Fundamentals of Group Representations 2134
- CHAPTER III. Abstract Theory of Representations of Finite Groups 3548
- 1 Orthogonality relations and the first fundamental relation 3649
- 2 Characters, class functions, and conjugacy classes 3952
- 3 One-dimensional representations 4255
- 4 The dimension theorem 4356
- 5 The theorem of Frobenius and Schur 4760
- Appendix to III. 5—Representations on real and quaternionic vector spaces 5063
- 6 Representations and group structure 5568
- 7 Projections in the group algebra 5669
- 8 Fourier analysis 5770
- 9 Direct products 5972
- 10 Restrictions 5972
- 11 Subgroups of index 2 6073
- 12 Examples 6275

- CHAPTER IV. Representations of Concrete Finite Groups. I: Abelian and Clifford Groups 6578
- CHAPTER V. Representations of Concrete Finite Groups. II: Semidirect Products and Induced Representations 7790
- CHAPTER VI. Representations of Concrete Finite Groups. Ill: The Symmetric Groups 95108
- CHAPTER VII. Compact Groups 121134
- 1 C∞-manifolds: A review 121134
- 2 Lie groups and Lie algebras 128141
- 3 Haar measure on Lie groups 133146
- 4 Matrix groups 135148
- 5 The classical groups 137150
- 6 Homotopy and covering groups 146159
- 7 Spin groups 152165
- 8 The structure of compact groups 155168
- 9 Representations of compact groups: Abstract theory 155168
- 10 The Peter-Weyl theorem 158171

- CHAPTER VIII. The Structure of Compact Semisimple Groups 165178
- 1 Maximal tori 165178
- 2 The Killing form 170183
- 3 Representations of tori 173186
- 4 Representations of SU(2) and sl(2,C) 174187
- 5 Roots and root spaces 177190
- 6 Fundamental systems and their classification 183196
- 7 Regular and singular elements 189202
- 8 The Weyl group 192205
- 9 The classical groups 196209

- CHAPTER IX. The Representations of Compact Semisimple Groups 205218
- 1 Geometry of the Cartan-Stiefel diagram 206219
- 2 The geometry of integral forms 210223
- 3 The Weyl integration formula 213226
- 4 Maximal weights 215228
- 5 The classification theorem and the Weyl character formula 217230
- 6 Consequences of the Weyl character formula 219232
- 7 Representation theory: The algebraic approach 225238
- 8 Representations of the classical groups 227240
- 9 Determinant formulas for the classical characters 237250
- 10 Real and quaternionic representations of the classical groups 242255
- 11 Tensors, permutations, and the Frobenius character formula 246259

- Appendices 253266
- Bibliography 261274
- Index 263276
- Back Cover Back Cover1280