Semigroups, rings, and Markov chains

Details Semigroups, rings, and Markov chains Semigroups, rings, and Markov chains

2000-06-20

arxiv.org/abs/math/0006145v1

Mathematics

Probability

To appear in J. Theoret. Probab

Scientific paper

We analyze random walks on a class of semigroups called ``left-regular bands''. These walks include the hyperplane chamber walks of Bidigare, 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.

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