Representations of Groups: A Computational Approach

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Overview

The representation theory of finite groups has seen rapid growth in recent years with the development of efficient algorithms and computer algebra systems. This is the first book to provide an introduction to the ordinary and modular representation theory of finite groups with special emphasis on the computational aspects of the subject. Evolving from courses taught at Aachen University, this well-paced text is ideal for graduate-level study. The authors provide over 200 exercises, both theoretical and computational, and include worked examples using the computer algebra system GAP. These make the abstract theory tangible and engage students in real hands-on work. GAP is freely available from www.gap-system.org and readers can download source code and solutions to selected exercises from the book's web page.

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Editorial Reviews

From the Publisher
"Representations of Groups: A Computational Approach, by Lux and Pahlings, is a well-constructed, dense, and, its introductory nature notwithstanding, pretty far-reaching text. Happily, it is also quite accessible....As it should be given what the authors have in mind, Representations of Groups: A Computational Approach is laden with examples and exercises, many of them supplied with hints. The book is well-written and anything but chatty and is based on the authors’ courses so that much of the material in the book has been field tested."
Michael Berg, MAA Reviews

"I enjoyed reading this book, and recommend it as a good modern second course in representation theory. The choice of topics and their arrangement are interesting, sometimes even surprising, but clearly well thought out."
Robert A. Wilson, Bulletin of the London Mathematical Society

"... this is really a wonderful, nice, and significant book, which is so well written, and it is recommended to a wide range of mathematicians, including not only graduate students but also many mathematicians of greater experience."
Shigeo Koshittani, Mathematical Reviews

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Product Details

Meet the Author

Klaus Lux is Associate Professor in the Department of Mathematics at the University of Arizona, Tucson.

Herbert Pahlings is Professor Emeritus at RWTH Aachen University in Germany.

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Table of Contents

Preface vii

List of frequently used symbols x

1 Representations and modules 1

1.1 Basic concepts 1

1.2 Permutation representations and G-sets 23

1.3 Simple modules, the "Meataxe" 36

1.4 Structure of algebras 50

1.5 Semisimple rings and modules 55

1.6 Direct sums and idempotents 63

1.7 Blocks 79

1.8 Changing coefficients 83

2 Characters 87

2.1 Characters and block idempotents 87

2.2 Character values 101

2.3 Character degrees 109

2.4 The Dixon-Schneider algorithm 113

2.5 Application - generation of groups 121

2.6 Character tables 135

2.7 Products of characters 139

2.8 Generalized characters and lattices 149

2.9 Invariant bilinear forms and the Schur index 164

2.10 Computing character tables - an example 173

3 Groups and subgroups 179

3.1 Restriction and fusion 179

3.2 Induced modules and characters 182

3.3 Symmetric groups 199

3.4 Permutation characters 206

3.5 Tables of marks 210

3.6 Clifford theory 223

3.7 Projective representations 238

3.8 Clifford matrices 259

3.9 M-groups 277

3.10 Brauer's induction theorem 281

4 Modular representations 289

4.1 p-modular systems 289

4.2 Brauer characters 301

4.3 p-projective characters 315

4.4 Characters in blocks 321

4.5 Basic sets 331

4.6 Defect groups 344

4.7 Brauer correspondence 350

4.8 Vertices 365

4.9 Green correspondence 378

4.10 Trivial source modules 386

4.11 Generalized decomposition numbers 396

4.12 Brauer's theory of blocks of defect one 407

4.13 Brauer characters of p-solvable groups 422

4.14 Some conjectures 426

References 443

Notation index 455

Subject index 457

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