
Linear Representations of Finite Groups
Übersetzt von Scott, Leonhard L.
Jean-Pierre Serre(Author)
Springer (Publisher)
Published on 1. September 1977
Book
Hardback
X, 172 pages
978-0-387-90190-9 (ISBN)
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.
Reviews / Votes
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)
More details
Series
Edition
1st ed. 1977. Corr. 5th printing 1996
Language
English
Place of publication
New York
United States
Target group
Primary & secondary/elementary & high school
Graduate
Edition type
Revised edition
Illustrations
X, 172 p.
Dimensions
Height: 240 mm
Width: 161 mm
Thickness: 16 mm
Weight
452 gr
ISBN-13
978-0-387-90190-9 (9780387901909)
DOI
10.1007/978-1-4684-9458-7
Schweitzer Classification
Other editions
Additional editions

Jean-Pierre Serre
Linear Representations of Finite Groups
Book
07/2012
Springer
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Content
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.