3 - Representations of Groups

Abstract

This chapter presents the fundamental theory of representations of finite groups, including modular representations. It discusses ordinary representations. By an ordinary representation of G, it means a representation over a field K of characteristic zero, and the character defined by it is called an ordinary character. Such a representation is always completely reducible and determined uniquely up to equivalence by the character. All representations considered are taken over the complex field C. In particular, every irreducible representation is absolutely irreducible. The chapter also reviews modular representation theory (p). By a (p-) modular representation, one means a representation of a group over a field of characteristic p > 0, and a character defined by such a representation is called a modular character. Some elementary facts on p-modular representations are established in the chapter for a fixed prime number p.