= Solution
<Schur's lemma> states that an <intertwiner> $T:V\to W$ between <irreducible representations> is either zero or an <isomorphism>. In particular, inequivalent <irreducible representations> have no nonzero <intertwiners>. For a finite-dimensional <irreducible representation> over the <complex numbers>, every commuting <endomorphism> is scalar:
$$
\boxed{TR(g)=R(g)T\ \text{for all }g\quad\Longrightarrow\quad T=\lambda I.}
$$
Indeed, the <kernel> and <image> of an <intertwiner> are <invariant subspaces>, proving the first statement. For the second, choose an <eigenvalue> $\lambda$ of $T$. The <intertwiner> $T-\lambda I$ has nonzero <kernel>, so it must vanish. The scalar conclusion uses an algebraically closed coefficient <field>: over the real numbers, planar rotation through $\pi/2$ gives an <irreducible> real <representation> of $\mathbb Z_4$ and commutes with itself without being a real scalar <matrix>.
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