Some properties for morphism of representations

Abstract

Let $\psi :G\to GL(V)$ and $\phi :G\to GL(W)$ be representations of finite group $G$. A linear map $T:V\to W$ is called a morphism from $\psi$ to $\phi$ if it satisfys $T{\psi }_g={\phi }_{g}T$ for each $g\in G$ and let ${\mathrm{Hom}}_G(\psi ,\phi )$ denote the set of all morphisms. In this paper, we make full stufy of the subspace ${\mathrm{Hom}}_G(\psi ,\phi )$. As byproducts, we include the proof of the first orthogonality relation and Schur's orthogonality relation.