# Tensor product between a representation and its dual

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?

• Show that $\chi\chi^*$ always has the trivial character as a constituent. – Lord Shark the Unknown Apr 9 at 10:12
• So, since $| \chi(1) |^{2} >1$ and $(\chi \chi^{*},\chi_{triv})_{G} = \dfrac{\sum_{g \in G} | \chi(g)|^{2}}{|G|} \in \mathbb{N}$ then $(\chi \chi^{*},\chi_{triv})_{G} >= 1$? – Davide Motta Apr 9 at 10:27
• Exactly${}{}{{}}$! – Lord Shark the Unknown Apr 9 at 10:45
• Easier than it looks, thanks – Davide Motta Apr 9 at 11:17

Note that $$M\otimes M^*\cong \mathrm{End(M)}$$. Can you spot any submodules?
• I thought $<v_i \otimes v_i^{*}>, \, i=1, \dots , \chi(1)$ but i don’t know how to prove it is G-stable – Davide Motta Apr 9 at 9:46
• Here it is shown that under the isomorphism $M\otimes M^*\cong \mathrm{End}(M)$, $G$ acts on $\mathrm{End}(M)$ by $f\mapsto gfg^{-1}$. From this description you can easily find a trivial submodule. – Alvaro Martinez Apr 9 at 15:36