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. If I'm not horribly mistaken, I think I can prove the result if $V$ is irreducible and nontrivial and $G$ has nontrivial center, but I'm not sure I can assume this.

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.

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.

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. If I'm not horribly mistaken, I think I can prove the result if $V$ is irreducible and nontrivial and $G$ has nontrivial center, but I'm not sure I can assume this.