I am looking for alternative/simpler expressions for the sum
$F_{\mu_1,\mu_2,\mu_3}(h,k)\equiv\frac{1}{|G|}\sum_{g\in G} \chi_{\mu_1}(g)\chi_{\mu_2}(g h)\chi_{\mu_3}(g k)$
where $g,h,k\in G$ and $\chi_{\mu_{1,2,3}}$ are irreducible characters over $\mathbb{C}$ of finite group $G$. I can see that there are general constraints on this quantity (e.g., it vanishes unless the product of irreps $V_{\mu_1} \otimes V_{\mu_2} \otimes V_{\mu_3}$ contains the trivial representation). I can also write down a cumbersome expression in terms of intertwiners. Does a simpler expression exist? If nothing more can be said in general, what about the specific case $G=S_n$, the symmetric group on $n$ elements?
