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

TL;DR

This paper investigates the eigenvalue distribution of symmetric group permutation representations on k-tuples under Ewens distribution, revealing the limiting point process and eigenvalue gap probabilities as the group size grows large.

Contribution

It introduces the limiting eigenvalue point process for symmetric group representations on k-tuples and provides a formula for eigenvalue gap probabilities, including a combinatorial method for coefficient calculation.

Findings

Derived the limiting eigenvalue point process as n approaches infinity.

Provided a formula for eigenvalue gap probabilities in the microscopic regime.

Developed a combinatorial procedure for computing series coefficients.

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.