Let $\rho : S\_n \rightarrow \text{GL}(n, \mathbb{C})$ be the homomorphism mapping a permutation $g$ to its permutation matrix. Let $\chi(g) = \text{Trace}(\rho(g))$.


What is the value of $\langle \chi, \chi \rangle = \displaystyle \frac{1}{n!} \sum\_{g \in S\_n} \chi(g)^2$ ? Computing this expression for small $n$ yields $2$. Is this always true?



Permutation representation inner product
You are computing the inner product of $\chi$ with itself. 


Since $\chi=\mathrm{triv}+\mathrm{std}$ as a $S\_n$-module, with $\mathrm{triv}$ being the trivial reprsentation, and $\mathrm{std}$ its orthogonal complement, which is an irreducible $S\_n$-module, and since the inner product is, well, an inner product and distinct irreducible characters are orthogonal, your $2$ follows from $$\langle\chi,\chi\rangle=\langle\mathrm{triv}+\mathrm{std},\mathrm{triv}+\mathrm{std}\rangle=\langle\mathrm{triv},\mathrm{triv}\rangle+\langle\mathrm{std},\mathrm{std}\rangle=1+1$$.


