= Trivial representation in a tensor square
{title2=$\dim(V\otimes V)^G=\dim\operatorname{Hom}_G(V^*,V)$}
For a finite-dimensional irreducible complex representation $V$, invariant tensors in $V\otimes V$ correspond to equivariant maps $V^*\to V$. <Schur's lemma> makes their dimension one if $V$ is self-dual and zero otherwise. In particular a real-character irreducible representation has precisely one trivial copy in its tensor square. For arbitrary complex irreducibles, the unconditional statement is that $V\otimes V^*$ contains the trivial representation once.
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