Some properties for morphism of representations

TL;DR

This paper studies the properties of morphisms between group representations, providing a detailed analysis of the Hom space and including proofs of fundamental orthogonality relations in representation theory.

Contribution

It offers a comprehensive study of the subspace of morphisms between representations and includes proofs of key orthogonality relations, enhancing understanding of representation morphisms.

Findings

Analysis of the structure of Hom_G(ψ, φ)

Proof of the first orthogonality relation

Proof of Schur's orthogonality relation

Abstract

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