Preface

The first chapter is devoted to tensor products: This basic and fundamental notion is hardly taught at undergraduate level, and I want the reader to be immediately familiar with it, once and for all. • Chapter 2 is a general introduction to group representations (using a bit of categorical language). In particular, we treat the case of representations on sets, their classification, as well as Burnside marks, and we also introduce the reader to general linear representations, their language, basic facts such as Schur’s lemma, and common examples and counterexamples. • Chapter 3 contains the general results about characteristic zero representations, without any further assumption about the ground field (hence, the ground field may be Q). For example, it provides a proof of the general “Fourier inversion formula” in this general context, as well as the more classical results about the Galois action on conjugacy classes in relation to the number of irreducible characters. One paragraph is devoted to what was, for Frobenius, the origin of the whole character theory: the group determinant. • Chapter 4 “plays around with the ground ring.” It contains a description, free of the theory of central simple algebras, of the integers which constitute the degree of an irreducible character: Character theory is sufficient to establish, for example, that the dimension of the occurring skewfields over their center is a square. It also contains a short introduction to reflection groups. • Chapter 5 is devoted to induction–restriction. The first part has no assumption about the characteristic of the field, insisting on the formal equalities and isomorphisms between occurring bimodules or functors. The second part deals, more classically, with the setting of characteristic zero ground fields and class functions. • The first part of Chap. 6 contains a proof of Brauer’s characterization of characters, while its second part contains some (less classical in textbooks) applications to “subgroups controlling ...-fusion” and normal ...-complements. As an application, it also contains a treatment of Frobenius groups. • In order to study representations of finite groups on graded modules, Chap. 7 starts with a short introduction to basic notions concerning graded modules and graded algebras. In the second part, we introduce the notion of graded characters with applications to generalization of Molien’s Formula. As a particular case, we study the action of G on SðVÞ (the symmetric algebra of V) if G acts on a vector space V . We conclude by an introduction to the main characterization of reflection groups. • The last chapter is devoted to the study of the Drinfeld double of a finite group. We start by a quick introduction to Hopf algebras and their algebra representations, then we introduce the notion of universal R-matrix and show that the representation category of the Drinfeld double is a ribbon category. We conclude by the definition of the S-matrix of that ribbon category, and by explicitly computing the associated representation of GL2ðZÞ. viii Preface