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Linear Representations of Finite Groups

  • Textbook
  • © 1977

Overview

Authors:
  1. Jean-Pierre Serre

    1. Chaire d’algèbre et géométrie, Collège de France, Paris, France

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

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Table of contents (19 chapters)

  1. Front Matter

    Pages i-x
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  2. Representations and Characters

    1. Front Matter

      Pages 1-1
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    2. Generalities on linear representations

      • Jean-Pierre Serre
      Pages 3-9

    3. Character theory

      • Jean-Pierre Serre
      Pages 10-24

    4. Subgroups, products, induced representations

      • Jean-Pierre Serre
      Pages 25-31

    5. Compact groups

      • Jean-Pierre Serre
      Pages 32-34

    6. Examples

      • Jean-Pierre Serre
      Pages 35-43

  3. Representations in Characteristic Zero

    1. Front Matter

      Pages 45-45
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    2. The group algebra

      • Jean-Pierre Serre
      Pages 47-53

    3. Induced representations; Mackey’s criterion

      • Jean-Pierre Serre
      Pages 54-60

    4. Examples of induced representations

      • Jean-Pierre Serre
      Pages 61-67

    5. Artin’s theorem

      • Jean-Pierre Serre
      Pages 68-73

    6. A theorem of Brauer

      • Jean-Pierre Serre
      Pages 74-80

    7. Applications of Brauer’s theorem

      • Jean-Pierre Serre
      Pages 81-89

    8. Rationality questions

      • Jean-Pierre Serre
      Pages 90-101

    9. Rationality questions: examples

      • Jean-Pierre Serre
      Pages 102-110

  4. Introduction to Brauer Theory

    1. Front Matter

      Pages 113-113
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    2. The groups RK(G), R k (G), and P k (G)

      • Jean-Pierre Serre
      Pages 115-123

    3. The cde triangle

      • Jean-Pierre Serre
      Pages 124-130

    4. Theorems

      • Jean-Pierre Serre
      Pages 131-137
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Keywords

About this book

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.

Reviews

From the reviews:

"Serre’s book gives a fine introduction to representations for various audiences . . . As always with Serre, the exposition is clear and elegant, and the exercises contain a great deal of valuable information that is otherwise hard to find . . . it is highly recommended for specialists and nonspecialists alike." (Bulletin Of The American Mathematical Society)

Authors and Affiliations

  • Chaire d’algèbre et géométrie, Collège de France, Paris, France

    Jean-Pierre Serre

Bibliographic Information

  • Book Title: Linear Representations of Finite Groups

  • Authors: Jean-Pierre Serre

  • Series Title: Graduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4684-9458-7

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1977

  • Hardcover ISBN: 978-0-387-90190-9Published: 01 September 1977

  • Softcover ISBN: 978-1-4684-9460-0Published: 11 July 2012

  • eBook ISBN: 978-1-4684-9458-7Published: 06 December 2012

  • Series ISSN: 0072-5285

  • Series E-ISSN: 2197-5612

  • Edition Number: 1

  • Number of Pages: X, 172

  • Additional Information: Title of the original French edition: Representations lineaires des groupes finis

  • Topics: Group Theory and Generalizations

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