Eigenvectors for a random walk on a left-regular band

We present a simple construction of the eigenvectors for the transition matrices of random walks on a class of semigroups called left-regular bands. These walks were introduced and analyzed by Brown, and they include the hyperplane chamber walks of Bidigare, Hanlon and Rockmore. This construction leads to new concise proofs of several of the known results about these walks. We also explain how tools from poset topology can be used to extract an eigenbasis for the transition matrices of the hyperplane chamber walks, and indicate the connection with a method recently described by Denham.