Schur's Lemma

Statement

Let $G$ be any group and let $V$ and $W$ be irreducible representations. Then

1. A morphism $V\to W$ is either zero or an isomorphism

2. Every morphism $f:V\to V$ has the form $f(v)=\lambda v$ for some $\lambda \in \mathbb{C}$ (or other underlying field.)

$${\dim }_{\mathbb{C}}{\text{Hom}}_G(V,W)=\{\begin{array}{ll}1& V\cong W\\ 0& V≆W\end{array}$$

Proof

todo … Haha… maybe someday.