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 characยญ 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
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; Mackeyโs criterion -- 8 Examples of induced representations -- 9 Artinโs theorem -- 10 A theorem of Brauer -- 11 Applications of Brauerโs 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
