I have asked this in math.SE (<https://math.stackexchange.com/questions/2698772/factorizations-in-terms-of-characters>) but it was barely viewed.


I have seen mention in different places that the number of solutions to the factorization equation $\pi\_1\cdots \pi\_r=1$ in $S\_n$, is given by
$$ \frac{|C\_{\lambda\_1}|\cdots |C\_{\lambda\_r}|}{n!}\sum\_{\mu\vdash n} \frac{\chi\_\mu(\lambda\_1)\cdots \chi\_\mu(\lambda\_r)}{(\chi\_\mu(1))^{r-2}},$$ where $\lambda\_i$ is the cycle type of $\pi\_i$ and $\chi$ are the characters.


Where can I find a nice exposition of this topic?



Factorizations in terms of characters
Your formula is a simple consequence of the fact that for any finite group $G$, the elements $\frac{\chi(1)}{|G|}\sum\_{w\in G}\chi(w^{-1})w$ form a set of orthogonal idempotents for the center of the group algebra $\mathbb{C}G$, where $\chi$ ranges over all irreducible characters of $G$. The proof uses only the orthogonality relations for irreducible characters. See for instance the beginning of Section 3 of <http://math.ubbcluj.ro/~olteanu/olteanu-mathematica.pdf>.


