= 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. For a <finite group>, 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.
For an arbitrary <group>, a real-valued <character> need not imply self-duality: the one-dimensional <group representation> $n\mapsto2^n$ of $\mathbb Z$ has real <character> but its <tensor square> has action $n\mapsto4^n$, so contains no <trivial representation>.
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