Schur--Weyl duality for diagonalizing a Markov chain on the hypercube

TL;DR

This paper applies algebraic combinatorics, including Schur--Weyl duality, to explicitly diagonalize a Markov chain on the hypercube, providing precise convergence rates.

Contribution

It introduces a novel algebraic approach to explicitly find eigenfunctions and convergence rates for a specific Markov chain on binary tuples.

Findings

Explicit orthonormal basis of eigenfunctions derived

Sharp convergence rates established

Algebraic combinatorics tools effectively applied

Abstract

We show how the tools of modern algebraic combinatorics -- representation theory, Murphy elements, and particularly Schur--Weyl duality -- can be used to give an explicit orthonormal basis of eigenfunctions for a "curiously slowly mixing Markov chain" on the space of binary $n$-tuples. The basis is used to give sharp rates of convergence to stationarity.