Let the permutation group on $n$ elements $S_n$ act on a set $S$ of size $k < n$ via permutations. Fix some ordering on the elements of $S$ to make this sensible.
Is there any way to understand the eigenvalues of the $k \times k$ matrices thus obtained?
Anything special happens to the answer of the above question if one looks at a subgroup $H < G$ which permutes these $k$ elements? Like anything can be said about the representation of $H$ thus obtained?