Statistical enumeration of groups by double cosets

Abstract

Let H and K be subgroups of a finite group G. Pick g∈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 GLn(Fq). 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 S2n, which leads to new applications of the Ewens's sampling formula of mathematical genetics. 3) Finally, if H and K are parabolic subgroups of Sn, the double cosets are ‘contingency tables’, studied by statisticians for the past 100 years.