Spectrum of random-to-random shuffling in the Hecke algebra

TL;DR

This paper extends the analysis of random-to-random shuffling from the symmetric group to the Hecke algebra, revealing eigenvalues as polynomials in q and simplifying prior proofs through novel algebraic connections.

Contribution

It introduces a generalized Markov chain on the Hecke algebra, computes its spectrum as polynomials in q, and simplifies existing proofs using advanced algebraic techniques.

Findings

Eigenvalues are polynomials in q with non-negative integer coefficients.

Setting q=1 recovers the spectrum for the symmetric group.

Provides a unified algebraic framework connecting to representation theory.

Abstract

We generalize random-to-random shuffling from a Markov chain on the symmetric group to one on the Type A Iwahori Hecke algebra, and show that its eigenvalues are polynomials in q with non-negative integer coefficients. Setting q=1 recovers results of Dieker and Saliola, whose computation of the spectrum of random-to-random in the symmetric group resolved a nearly 20 year old conjecture by Uyemura-Reyes. Our methods simplify their proofs by drawing novel connections to the Jucys-Murphy elements of the Hecke algebra, Young seminormal forms, and the Okounkov-Vershik approach to representation theory.