Eigenvectors for a random walk on a hyperplane arrangement

We find explicit eigenvectors for the transition matrix of the Bidigare–Hanlon–Rockmore random walk, from Bidigare et al. (1999) [1], . This is accomplished by using Brown and Diaconisʼ (1998) analysis in [3], of the stationary distribution, together with some combinatorics of functions on the face lattice of a hyperplane arrangement, due to Gelʼfand and Varchenko (1987) [10].