Let H and K be subgroups of a finite group G. This divides G into H-K double cosets. One may ask (1) how many double cosets are there? (2) what are their sizes? (e.g. pick g in G at random, what double coset is it likely to be in?) (3) Can one find 'nice names' for the double cosets? (4) what kind of algebra is the associated 'Hecke algebra' of bi-invariant functions? These questions are largely open, even if H=K is the Sylow-p subgroup of the symmetric group. There are some cases with answers; for parabolic subgroups of the symmetric group, the double cosets are 'contingency tables' studied by statisticians for 100 years. For finite groups of Lie type, the double cosets are indexed by the Weyl group and the induced distribution is the celebrated 'Ewens sampling formula' of genetics. The study illuminates the analysis of Gaussian elimination. I will try to tell the story 'in English' focusing on some tractible open problems.

This is joint work with Mackenzie Simper.