# Are such averages known with representations of $S_n$?

• Like is there a sense in which one can quantify that for two group elements (in different conjugacy classes) their characters are "close" for some fixed irreducible representation? (feel free to stick to $S_n$ if you need to)

• $\sum_{g \in S_n} \chi_\rho(g)=0$ by orthogonality with respect to the character of the identity representation. But are there results known when the $g$ is restricted to be in some subset of $S_n$ like may be a choice of generating set? (I am looking for some general expression)

• Or $\sum_{ \pi }\chi_{\pi} (g)$ where $\pi$ goes over all irreducible representations of $S_n$ for a fixed $g \in S_n$. Are anything like these done or known to be doable?

The third question asks what can be said about $\sum \chi(g)$ for a fixed element $g \in S_n$, where the sum runs over all irreducible characters of $S_n$. The answer is this: that sum is equal to the number of elements $x \in G$ such that $x^2 = g$. This follows by the Frobenius-Schur theorem together with the fact that the Frobenius-Schur indicator of every irreducible character of a symmetric group is $+1$.