Semigroup and Ring Theoretical Methods in Probability

This is an expanded version of a series of four lectures designed to show algebraists how ring theoretical methods can be used to analyze an interesting family of finite Markov chains. The chains happen to be random walks on semigroups, and the analysis is based on a study of the associated semigroup algebras. The paper is divided into four sections, which correspond roughly to the four lectures: 1. Examples. 2. Semigroup interpretation. 3. Algebraic analysis. 4. Connections with Solomon’s descent algebra. Two appendices give a self-contained treatment of bands (also called idempotent semigroups) and their linear representations. The work described here grew out of my joint work with Persi Diaconis [5]. It is a pleasure to thank him for many stimulating conversations.