Properties of characters of representations

Question

Suppose $\chi$ is the character of the representation $\sigma:G\rightarrow \mathbb{C}$ (where $G$ could be assumed to be the symmetric group as well), then do we know anything about the following sum $$\sum_{\sigma}\sum_{x,y \in G} \chi_\sigma(x)^*\chi_\sigma(y)$$ where the outer sum over $\sigma$ runs over all possible irreducible representations of $G$.

Answer 1

Note that $\chi_\sigma^*=\overline{\chi_\sigma}$. Recall the orthogonality of columns in the character table of $G$: $$ \sum_{\sigma}\overline{\chi_\sigma}(g_i)\chi_\sigma(g_j)=|Z_G(g_i)|\delta_{ij}, $$ where the set $\{g_i\}$ is a set of representatives for the conjugacy classes in $G$ and $Z_G(g_i)$ is the centralizer of $g_i$. Suppose there are $r$ conjugacy classes, hence $r$ irreducible representations up to isomorphism. Therefore your sum is $$ \sum_{k=1}^r\sum_{x,y\in G}\overline{\chi_{\sigma_k}(x)}\chi_{\sigma_k}(y)= \sum_{k=1}^r\sum_{i,j=1}^r\frac{|G|}{|Z_G(g_i)||Z_G(g_j)|}\overline{\chi_{\sigma_k}(g_i)}\chi_{\sigma_k}(g_j)= \sum_{i,j=1}^r\sum_{k=1}^r\frac{|G|}{|Z_G(g_i)||Z_G(g_j)|}\overline{\chi_{\sigma_k}(g_i)}\chi_{\sigma_k}(g_j). $$ Now, by the orthogonality relation, the sum over $k$ is zero unless $i=j$, in which case it's equal to the size of the centralizer of $g_i$. Thus we have that your sum is equal to $$ \sum_{i=1}^r\frac{|G|}{|Z_G(g_i)|^2}|Z_G(g_i)|=\sum_{i=1}^r\frac{|G|}{|Z_G(g_i)|}=|G|. $$