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arXiv · 1901.06721

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

Benjamin Tsou
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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.

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Benjamin Tsou. 2019-01-20. The Eigenvalue Point Process for Symmetric Group Permutation Representations on $k$-tuples. https://arxiv.org/abs/1901.06721

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