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I have the following question:

Let $G$ be a finite group, $\hat{G}$ the set of irreducible characters of $G$ (up to equivalence), and $n = |G|$. Given $x,g \in G,$ show \begin{equation*} \frac{1}{n}\sum_{\chi \in \hat{G}}\mathrm{deg}(\pi)\chi_{\pi}(gx^{-1})=\delta_{g,x}. \end{equation*}

I can prove that the sum is equal to $1$ when $g=x$, but I haven't been able to make progress for the other case.

I have tried rearranging the sum, and also introducing a factor of $\chi'(gx^{-1})$ for a suitable $\chi'$ in the hopes of showing it doesn't change the sum, however, with $\hat{G}$ not being abelian this wasn't fruitful.

All answers are appreciated, hints are preferred.

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It is well known that the regular representation $(\rho, V)$ of $G$ decomposes as $$\rho\sim\oplus_{\pi}\deg(\pi)\pi,$$ where the direct sum is taken over a complete set of inequivalent irreducible representations of $G$ (for instance, see here). Therefore, we can write $\chi_\rho=\sum_\pi\deg(\pi)\chi_\pi$. In other words, $$\chi_\rho(g)=\sum_\pi\deg(\pi)\chi_{\pi}(g)$$ for all $g\in G$.

We have $$\chi_{\rho}(g)=\begin{cases}n,&g=1\\0,&g\neq 1\end{cases}$$ by direct calculations. Thus $$\sum_\pi\deg(\pi)\chi_{\pi}(gx^{-1})=\chi_\rho(gx^{-1})=\begin{cases}n,&g=x\\0,&g\neq x\end{cases}.$$

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  • $\begingroup$ Can you briefly elaborate on the direct calculations mentioned to compute $\chi_{\rho}(g)$? $\endgroup$
    – carraig
    Commented Oct 21, 2021 at 18:14
  • $\begingroup$ @carraig: Given $g\in G$, $\rho_g$ takes a basis element $h$ to $gh$. If $g\neq 1$, then $h\neq gh$. This implies that the diagonal entries of the matrix corresponding to $\rho_g$ are zero. $\endgroup$
    – cqfd
    Commented Oct 22, 2021 at 3:58

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