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Linear Representations of Finite Groups

  • Jean-Pierre Serre

Part of the Graduate Texts in Mathematics book series (GTM, volume 42)

Table of contents

  1. Front Matter
    Pages i-x
  2. Representations and Characters

    1. Front Matter
      Pages 1-1
    2. Jean-Pierre Serre
      Pages 3-9
    3. Jean-Pierre Serre
      Pages 10-24
    4. Jean-Pierre Serre
      Pages 25-31
    5. Jean-Pierre Serre
      Pages 32-34
    6. Jean-Pierre Serre
      Pages 35-43
  3. Representations in Characteristic Zero

    1. Front Matter
      Pages 45-45
    2. Jean-Pierre Serre
      Pages 47-53
    3. Jean-Pierre Serre
      Pages 54-60
    4. Jean-Pierre Serre
      Pages 61-67
    5. Jean-Pierre Serre
      Pages 68-73
    6. Jean-Pierre Serre
      Pages 74-80
    7. Jean-Pierre Serre
      Pages 81-89
    8. Jean-Pierre Serre
      Pages 90-101
    9. Jean-Pierre Serre
      Pages 102-110
  4. Introduction to Brauer Theory

    1. Front Matter
      Pages 113-113
    2. Jean-Pierre Serre
      Pages 115-123
    3. Jean-Pierre Serre
      Pages 124-130
    4. Jean-Pierre Serre
      Pages 131-137
    5. Jean-Pierre Serre
      Pages 138-146
    6. Jean-Pierre Serre
      Pages 147-158
    7. Jean-Pierre Serre
      Pages 159-162
  5. Back Matter
    Pages 167-173

About this book

Introduction

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.

Keywords

Darstellung (Math.) Endliche Gruppe Finite algebra character theory mathematics proof theorem

Authors and affiliations

  • Jean-Pierre Serre
    • 1
  1. 1.Chaire d’algèbre et géométrieCollège de FranceParisFrance

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4684-9458-7
  • Copyright Information Springer Science+Business Media New York 1977
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4684-9460-0
  • Online ISBN 978-1-4684-9458-7
  • Series Print ISSN 0072-5285
  • Series Online ISSN 2197-5612
  • Buy this book on publisher's site