Richard P. Stanley Seminar in Combinatorics: Spectra of shuffling operators as evaluations of supersymmetric functions

In 2014, Reiner, Saliola, and Welker introduced the k-random-to-random shuffles, a family of Markov chains with two mysterious properties: (1) its members pairwise commute, and (2) the eigenvalues of every member (after normalization) are non-negative integers. Yet little was understood about why these properties hold; for example, there were no known closed-form expressions for the eigenvalues.
In this talk, I will discuss recent work with Patty Commins, Darij Grinberg, and Franco Saliola that aims to address this “why.” First, I will show that the eigenvalues of the k-random-to-random shuffles can be expressed as evaluations of certain supersymmetric functions. Second, I will explain how q-deforming these shuffles in the Iwahori–Hecke algebra sheds light on their commutativity; in this setting, the eigenvalues become polynomials in q with non-negative integer coefficients.

For information about the Richard P. Stanley Seminar in Combinatorics, visit… https://math.mit.edu/combin/