Semigroups , Rings , and Markov

We analyze random walks on a class of semigroups called \left-regular bands". These walks include the hyperplane chamber walks of Bidi-gare, Hanlon, and Rockmore. Using methods of ring theory, we show that the transition matrices are diagonalizable and we calculate the eigenvalues and multiplicities. The methods lead to explicit formulas for the projections onto the eigenspaces. As examples of these semigroup walks, we construct a random walk on the maximal chains of any distributive lattice, as well as two random walks associated with any matroid. The examples include a q-analogue of the Tsetlin library. The multiplicities of the eigenvalues in the matroid walks are \generalized derangement numbers", which may be of independent interest. 1. Introduction There are many tools available for the study of random walks on nite groups, an important one being representation theory 14]. For nite semigroups, on the other hand, there is no representation theory comparable to that for groups. And, although there is some general theory of random walks 21, 16], much less is known for semigroups than for groups. We consider here a special class of nite semi-groups whose irreducible representations can be worked out explcitly (they are all 1-dimensional), and we use this information to analyze the random walks. In particular, we calculate the eigenvalues, which turn out to be real. The semigroups we treat are called \left-regular bands" in the semigroup literature. There are many interesting examples of them, including the hyperplane chamber walks introduced by Bidigare, Hanlon, and Rockmore 5], as well as several new examples. Our approach via representation theory provides a clear conceptual explanation for some of the remarkable features of the hyperplane chamber walks proved in 5, 10].