# On an equation satisfied by the irreducible characters of a finite group

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.

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 $$\sum_\pi\deg(\pi)\chi_{\pi}(gx^{-1})=\chi_\rho(gx^{-1})=\begin{cases}n,&g=x\\0,&g\neq x\end{cases}.$$
• Can you briefly elaborate on the direct calculations mentioned to compute $\chi_{\rho}(g)$? Commented Oct 21, 2021 at 18:14
• @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.