On characters of finite groups (Q1675126)

Among several books on character theory of finite groups, one should mention [\textit{I. M. Isaacs}, Character theory of finite groups. New York-San Francisco-London: Academic Press, a subsidiary of Harcourt Brace Jovanovich, Publishers (1976; Zbl 0337.20005)] and [\textit{B. Huppert}, Character theory of finite groups. Berlin: Walter de Gruyter (1998; Zbl 0932.20007)] in which rudiments of character theory of finite groups over the complex field, ordinary character theory, is well-developed and both are used as text books. The present book by M. Broue although covers all the topics that a student should learn to do research on character theory but uses a different approach that is the language of category theory at several stages of the book. In general let \(\mathcal{C}\) be a category, a representation of a group \(G\) on \(\mathcal{C}\) is a pair \((X,\rho)\), where \(X\) is an object of \(\mathcal{C}\) and \(\rho\) is a group morphism from \(G\) into the group \(\Aut X\) of automorphisms of \(X\). Now, if we choose \(\mathcal{C}\) to be the category \(\mathrm{vect}_{k}\) of finite dimensional vector spaces over the field \(k\), then \(\rho\) is the usual representation of the group \(G\). Characteristic zero representations are dealt with in the longest chapter of the book which is Chapter 3 of the book through which fundamental theorems of character theory are proved and as a consequence the famous Burnside \((p,q)\)-theorem is proved. The construction of character tables is part of this chapter. Chapter 7 of the book is dedicated to the graded representations and applications of polynomial invariants of finite groups. In the final chapter, the author addresses the more recent notion of the Drinfeld double of a finite group and representation of \(\mathrm{GL}_{2}(\mathbb{Z})\). The book can serve as a textbook for a course on characters of finite groups at graduate level. As a prerequisite, students require knowledge of the tensor product of vector spaces and modules, which is provided in Chapter 1 of the book. The reader should familiarize himself with category theory and functors to enjoy most of materials covered in the book. Each chapter of the book contains exercises which help better understanding the concept covered in that chapter. This book contains more than what is needed for a course on character theory for graduate students, but added materials will enhance the research of the readers of the book. Although there is no mention of the irreducible representations of the symmetric and alternating groups in the book, this does not effect the comprehensiveness of the book. The book is written by one of the best character theorists, and he has done a great job. I am sure, teachers and students interested in character theory both enjoy reading this book.