= Solution
<Schur lemma> over $\mathbb C$ states that an <intertwining operator> $T:V\to W$ between finite-dimensional irreducible complex representations is either zero or an isomorphism. In particular, every intertwining endomorphism of $V$ is a scalar multiple of the identity.
Indeed, $\ker T\subseteq V$ and $\operatorname{im}T\subseteq W$ are <invariant subspaces>. If $T\ne0$, irreducibility gives $\ker T=0$ and $\operatorname{im}T=W$, so $T$ is an isomorphism. For an endomorphism $T:V\to V$, the <fundamental theorem of algebra> gives an eigenvalue $\lambda\in\mathbb C$. Then $T-\lambda I$ is an intertwiner with nonzero kernel, so the first part forces $T-\lambda I=0$. Hence $T=\lambda I$, completing the <Proof of Schur lemma>.
Back to article page
