TITLE: For a representation V of a finite group G, when is Hom(W, W⊗V) trivial for all irreps W?
QUESTION [6 upvotes]: This is probably really easy, but I just need someone to help me get mentally unstuck.  As part of a description of the McKay correspondence, I want to show that if $G$ is a finite subgroup of $SU(2)$ and $V$ the corresponding 2-dimensional representation, then $\dim \text{Hom}(W, W \otimes V) = 0$ for any irreducible representation $W$ of $G$.  I suspect the result is true in slightly greater generality, but it clearly can't always be true.  Since $\dim \text{Hom}(W, W \otimes V) = \dim \text{Hom}(W \otimes W^{\ast}, V)$, the result is false if, for example, $V \simeq W \otimes W^{\ast}$ or is a direct summand thereof for some $W$.
So I am wondering when, for a given $G$ and $V$, it is always true that $\dim \text{Hom}(W, W \otimes V) = 0$ for all irreducible representations $W$.  One can easily reduce to the case that $V$ is irreducible.

REPLY [2 votes]: For a compact group $G$ one can define the following equivalence: given  two irreps $X$ and $Y$,  $X \sim Y$  if they both appear as summands in  a finite string of tensor products of irreps $X_1 \otimes X_2 \otimes \dots X_n$. The equivalence classes have the structure of an abelian group which turns out to be the dual of the centre of $G$. This was conjectured in  http://arXiv.org/abs/math/0311170 and proven in http://arxiv.org/abs/math/0312257.
Thus $Hom(W \otimes W^*,V) \neq 0$ iff V is in the identity class (i.e. the centre acts trivially on $V$).