Trivial representation in a tensor square ($\dim (V\otimes V{)}^G=\dim {\mathrm{Hom}}_G(V^{\ast },V)$)

For a finite-dimensional irreducible complex representation $V$, invariant tensors in $V\otimes V$ correspond to equivariant maps $V^{\ast }\to V$. 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 $V\otimes V^{\ast }$ contains the trivial representation once.