Combinatorics Seminar

Classically, the cycle index of a permutation group is a polynomial developed to count colorings modulo an action of the group, often known as the Pólya enumeration theorem. The generating function whose n-th coefficient is given by the cycle index of the full symmetric group of n letters has an interesting factorization property. In this talk, we will dive into the structure behind this factorization, which is also present for n x n matrices over the finite field of p elements. As an application, we will generalize two results of Friedman and Washington on the asymptotic distribution of the cokernal of a random n x n integral p-adic matrix with respect to the Haar measure, when n goes to infinity. The limiting distribution follows a generalized version of the Cohen-Lenstra distribution.

This is joint work with Yifeng Huang. Speaker(s): Gilyoung Cheong (University of Michigan)