Character sum involving conjugacy classes

Question

I am tasked with showing

$$ \sum_{ x \in C_1, y \in C_2, z \in C_3} \chi(xyz) = \frac{|C_1| |C_2| |C_3| \chi(C_1)\chi(C_2)\chi(C_3)}{\chi(1)^2} $$

where $C_1,C_2,C_3$ are conjugacy classes in a finite group $G$, and $\chi$ is the character of an irreducible representation $\rho$.

A hint is that the left side is $\text{trace} \left( \sum_{ x \in C_1, y \in C_2, z \in C_3} \rho(xyz) \right)$. I can see by Schur's lemma this map acts by multiplication by a constant, but I don't see how to compute this constant.

Answer 1

Hint: Because $\rho$ is a homomorphism of groups we can decompose (as linear transformations) $$ \sum_{x\in C_1,y\in C_2,z\in C_3}\rho(xyz)=f_1\circ f_2\circ f_3, $$ where for all $i=1,2,3$, I denote the linear transformation $$ f_i=\sum_{x\in C_i}\rho(x). $$ Apply Schur's lemma also to individual transformations $f_1,f_2,f_3$.