On the Representations and Characters of Cat¹-Groups and Crossed Modules

Year 2019, Volume: 68 Issue: 1, 70 - 86, 01.02.2019

M. A. Dehghani B. Davvaz

https://doi.org/10.31801/cfsuasmas.443623

Cited By: 1

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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