If $\chi$ is the character of an irreducible representation of a finite group $G$ such that $\chi(1) > 1$, then I want to prove $\chi \chi^{*}$ is never irreducible.
My idea was to show $\sum_{g \in G}| \chi(g) |^4 > |G|$. However i can not find a suitable way following this idea.
Any hints?
Note that $M\otimes M^*\cong \mathrm{End(M)}$. Can you spot any submodules?