ON THE REPRESENTATIONS AND CHARACTERS OF CAT1 -GROUPS AND CROSSED MODULES

Abstract

Let G be a group and V a K-vector space. A K-linear representation of G with representation space V is a homomorphism : G ! GL(V ): The dimension of V is called the degree of . If is a representation of G, then the character is deÖned for g 2 G as g () = T r((g)). In this paper we study the representations and characters of cat1 -groups and crossed modules. We show that for class functions 1 and 2 of crossed module = (G; M; ; @), the inner product is Hermitian. Also, if = (G; M; ; @) is a Önite crossed module and is an irreducible character of , then X m2M; g2G (m; g) (m1 ; g1 ) = jGjjMj: Moreover, we present some examples of the character tables of crossed module.