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arXiv · 2102.04576

Statistical Enumeration of Groups by Double Cosets

Persi Diaconis·Mackenzie Simper
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Abstract

Let $H$ and $K$ be subgroups of a finite group $G$. Pick $g \in G$ uniformly at random. We study the distribution induced on double cosets. Three examples are treated in detail: 1) $H = K = $ the Borel subgroup in $GL_n(\mathbb{F}_q)$. This leads to new theorems for Mallows measure on permutations and new insights into the LU matrix factorization. 2) The double cosets of the hyperoctahedral group inside $S_{2n}$, which leads to new applications of the Ewens's sampling formula of mathematical genetics. 3) Finally, if $H$ and $K$ are parabolic subgroups of $S_n$, the double cosets are `contingency tables', studied by statisticians for the past 100 years.

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Persi Diaconis, Mackenzie Simper. 2021-02-08. Statistical Enumeration of Groups by Double Cosets. https://arxiv.org/abs/2102.04576

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