Statistical Enumeration of Groups by Double Cosets

TL;DR

This paper analyzes the distribution of double cosets in finite groups, providing new theoretical insights and applications across combinatorics, genetics, and statistics, through detailed examples involving matrix groups, permutation groups, and contingency tables.

Contribution

It introduces novel results on the distribution of double cosets with applications to Mallows measure, Ewens's sampling formula, and contingency tables, connecting group theory with diverse fields.

Findings

New theorems for Mallows measure on permutations

Applications of Ewens's sampling formula in genetics

Insights into contingency tables in statistics

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.