Linear Representations of Finite Groups

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 R_K(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

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.

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

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)

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

  Jean-Pierre Serre

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