@article{article_443623, title={On the Representations and Characters of Cat¹-Groups and Crossed Modules}, journal={Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics}, volume={68}, pages={70–86}, year={2019}, DOI={10.31801/cfsuasmas.443623}, author={Dehghani, M. A. and Davvaz, B.}, keywords={Cat¹-group,crossed module,chain complex,representation,character}, abstract={<div>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 </div> <div> <br> </div> <div> <span style="white-space:pre"> </span>∑_{m∈M,g∈G}ψ(m,g)ψ(m⁻¹,g⁻¹)=|G||M|. </div> <div> <br> </div> <div>Moreover, we present some examples of the character tables of crossed modules. </div>}, number={1}, publisher={Ankara University}