# Consequences of Schur's Lemma

Schur's Lemma states that given two irreducible representations $(\rho,V)$ and $(\pi,W)$ of a finite group $G$ (V and W are linear vector spaces over the same field), and a homomorphism $\phi\colon V\to W$ such that $\phi(\rho(g)\mathbf{v}) = \pi(\phi(\mathbf{v}))$ for all $g$ in $G$ and all $\mathbf{v}$ in $V$, then either $\phi$ is an isomorphism or $\phi(\mathbf{v}) = \mathbf{0}$ for all $\mathbf{v}$ in $V$. I understand that the first of these would imply that $V$ and $W$ are equivalent irreducible representations (meaning that $\rho$ and $\pi$ are related by a similarity transformation). But what is the consequence for $V$ and $W$ if $\phi(\mathbf{v}) = \mathbf{0}$ for all elements of $V$? A textbook I have read said that this indicates $V$ and $W$ are distinct irreducible representations. What does this mean? Does it mean that $V\bigcap W = \mathbf{0}?$ Or would it only mean that if $\rho$ and $\pi$ are identical maps?If so, how can I derive this consequence?

It doesn't imply anything about $V$ and $W$, just that $\phi$ is the zero map; there is always a zero map between any two representations.
The point of Schur's Lemma is that any non-zero map between irreducible representations is an isomorphism. So if you have a non-zero map $V\to W$, then it is an isomorphism, and $V$ and $W$ are equivalent. On the other hand, if $V$ and $W$ are not equivalent, then the only map $V\to W$ is the zero map. (Perhaps what your textbook is saying is that if all maps $V\to W$ are zero, then $V$ and $W$ are distinct; but the "all" is necessary).