Statistical Enumeration of Groups by Double Cosets

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 $G{L}_n({\mathbb{F}}_q)$. 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.

46 pages, 1 figure; minor edits