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Linear Representations of ­Finite Groups
v. 42 (Graduate Texts in Mathematics)

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Format
Hardback, 188 pages
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Paperback : £46.82

Published
United States, 30 October 1996

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.

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Product Description

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.

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Product Details
EAN
9780387901909
ISBN
0387901906
Other Information
X, 172 p.
Dimensions
24.2 x 16.1 x 1.7 centimeters (0.45 kg)

Table of Contents

I Representations and Characters.- 1 Generalities on linear representations.- 2 Character theory.- 3 Subgroups, products, induced representations.- 4 Compact groups.- 5 Examples.- Bibliography: Part I.- II Representations in Characteristic Zero.- 6 The group algebra.- 7 Induced representations; Mackey’s criterion.- 8 Examples of induced representations.- 9 Artin’s theorem.- 10 A theorem of Brauer.- 11 Applications of Brauer’s theorem.- 12 Rationality questions.- 13 Rationality questions: examples.- Bibliography: Part II.- III Introduction to Brauer Theory.- 14 The groups RK(G), Rk (G), and Pk(G).- 15 The cde triangle.- 16 Theorems.- 17 Proofs.- 18 Modular characters.- 19 Application to Artin representations.- Index of notation.- Index of terminology.

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Corrected fifth printing

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)

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