Solution
= Solution
Let $K=\ker\chi$, which is nontrivial. Every tensor power affording $\chi^n$ is trivial on $K$, so each of its irreducible constituents has $K$ in its kernel. Some irreducible representation is nontrivial on $K$; otherwise the regular representation, which contains every irreducible, would also be trivial on $K$. Choose its character $\psi$. Then $\psi$ is a constituent of no $\chi^n$, and
$$
\boxed{\langle\chi^n,\psi\rangle=0\quad(n\geq0)}.
$$