 Trivial representation in a tensor square

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. 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 $V\otimes V^{\ast }$ contains the trivial representation once.

For an arbitrary group, a real-valued character need not imply self-duality: the one-dimensional group representation $n\mapsto 2^n$ of $\mathbb{Z}$ has real character but its tensor square has action $n\mapsto 4^n$, so contains no trivial representation.