Linear Representations of Finite Groups

✍ Scribed by Jean-Pierre Serre (auth.)

• Publisher: Springer-Verlag New York
• Year: 1977
• Tongue: English
• Leaves: 179
• Series: Graduate Texts in Mathematics 42
• Edition: 1
• Category: Library

✦ Synopsis

This book consists of three parts, rather different in level and purpose: The first part was originally written for quantum chemists. It describes the correspondence, due to Frobenius, between linear representations and charac­ ters. This is a fundamental result, of constant use in mathematics as well as in quantum chemistry or physics. I have tried to give proofs as elementary as possible, using only the definition of a group and the rudiments of linear algebra. The examples (Chapter 5) have been chosen from those useful to chemists. The second part is a course given in 1966 to second-year students of I'Ecoie Normale. It completes the first on the following points: (a) degrees of representations and integrality properties of characters (Chapter 6); (b) induced representations, theorems of Artin and Brauer, and applications (Chapters 7-11); (c) rationality questions (Chapters 12 and 13). The methods used are those of linear algebra (in a wider sense than in the first part): group algebras, modules, noncommutative tensor products, semisimple algebras. The third part is an introduction to Brauer theory: passage from characteristic 0 to characteristic p (and conversely). I have freely used the language of abelian categories (projective modules, Grothendieck groups), which is well suited to this sort of question. The principal results are: (a) The fact that the decomposition homomorphism is surjective: all irreducible representations in characteristic p can be lifted "virtually" (i.e., in a suitable Grothendieck group) to characteristic O.

✦ Table of Contents

Front Matter....Pages i-x
Front Matter....Pages 1-1
Generalities on linear representations....Pages 3-9
Character theory....Pages 10-24
Subgroups, products, induced representations....Pages 25-31
Compact groups....Pages 32-34
Examples....Pages 35-43
Front Matter....Pages 45-45
The group algebra....Pages 47-53
Induced representations; Mackey’s criterion....Pages 54-60
Examples of induced representations....Pages 61-67
Artin’s theorem....Pages 68-73
A theorem of Brauer....Pages 74-80
Applications of Brauer’s theorem....Pages 81-89
Rationality questions....Pages 90-101
Rationality questions: examples....Pages 102-110
Front Matter....Pages 113-113
The groups R K (G), R k (G), and P k (G)....Pages 115-123
The cde triangle....Pages 124-130
Theorems....Pages 131-137
Proofs....Pages 138-146
Modular characters....Pages 147-158
Applications to Artin representations....Pages 159-162
Back Matter....Pages 167-172

✦ Subjects

Group Theory and Generalizations