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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?

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    $\begingroup$ Show that $\chi\chi^*$ always has the trivial character as a constituent. $\endgroup$ – Lord Shark the Unknown Apr 9 at 10:12
  • $\begingroup$ 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 $(\chi \chi^{*},\chi_{triv})_{G} >= 1$? $\endgroup$ – Davide Motta Apr 9 at 10:27
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    $\begingroup$ Exactly${}{}{{}}$! $\endgroup$ – Lord Shark the Unknown Apr 9 at 10:45
  • $\begingroup$ Easier than it looks, thanks $\endgroup$ – Davide Motta Apr 9 at 11:17
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Note that $M\otimes M^*\cong \mathrm{End(M)}$. Can you spot any submodules?

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  • $\begingroup$ I thought $<v_i \otimes v_i^{*}>, \, i=1, \dots , \chi(1)$ but i don’t know how to prove it is G-stable $\endgroup$ – Davide Motta Apr 9 at 9:46
  • $\begingroup$ 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. $\endgroup$ – Alvaro Martinez Apr 9 at 15:36

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