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

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.