The Eigenvalue Point Process for Symmetric Group Permutation Representations on $k$-tuples

Equip the symmetric group ${\mathfrak{S}}_n$ with the Ewens distribution. We study the eigenvalue point process of the permutation representation of ${\mathfrak{S}}_n$ on $k$-tuples of distinct integers chosen from the set $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n\}$. Taking $n\to \mathrm{\infty }$, we find the limiting point process in the microscopic regime, i.e. when the eigenvalue point process is viewed at the scale of the mean eigenvalue spacing. A formula for the limiting eigenvalue gap probability in an interval is also given. In certain cases, a power series representation exists and a combinatorial procedure is given for computing the coefficients.