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  • 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?

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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 $+1$.

For symmetric groups, this sum is always nonnegative, but that is not true for finite groups in general, although counterexamples are relatively scarce. An example where the sum is negative for some element is group number 56 of order 144 in the SmallGroups data base of Gap or Magma.

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