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For a compact group $G$ one can define the following equivalence: given two irreps $X$ and $Y$, $X \sim Y$ if they both compare 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 $V$).
For a compact group $G$ one can define the following equivalence: given two irreps $X$ and $Y$, $X \sim Y$ if they both compare 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 $V$).