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 φ:G→GL(V). The dimension of V is called the degree of φ. If φ is a representation of G, then the character φ is defined for g∈G as ψ_{g}(φ)=Tr(φ(g)). In this paper we study the representations and characters of cat¹-groups and crossed modules. We show that for class functions ψ₁ and ψ₂ of crossed module χ=(G,M,μ,∂), the inner product is Hermitian. Also, if χ=(G,M,μ,∂) is a finite crossed module and ψ is an irreducible character of χ, then

∑_{m∈M,g∈G}ψ(m,g)ψ(m⁻¹,g⁻¹)=|G||M|.

Moreover, we present some examples of the character tables of crossed modules.

Keywords

• Cat¹-group,crossed module,chain complex,representation,character

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