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

Abstract

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 $\{1,2,...,n\}$. Taking $n\to \infty$, 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.

43 pages

References in corpus (2)

• Vinogradov's mean value theorem via efficient congruencing
• Eigenvalue Fluctuations of Symmetric Group Permutation Representations on k-tuples and k-subsets