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

Authors : M A DEHGHANİ, B DAVVAZ

Pages : 70-86

Doi:10.31801/cfsuasmas.443623

View : 25 | Download : 13

Publication Date : 2019-02-01

Article Type : Research Paper

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→GLinsert ignore into journalissuearticles values(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}insert ignore into journalissuearticles values(φ);=Trinsert ignore into journalissuearticles values(φinsert ignore into journalissuearticles values(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=insert ignore into journalissuearticles values(G,M,μ,∂);, the inner product is Hermitian. Also, if χ=insert ignore into journalissuearticles values(G,M,μ,∂); is a finite crossed module and ψ is an irreducible character of χ, then ∑_{m∈M,g∈G}ψinsert ignore into journalissuearticles values(m,g);ψinsert ignore into journalissuearticles values(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