Combinatorial Topology, Random Walks and Global Dimension
Abstract: In recent years people like Bidigare, Hanlon and Rockmore, Brown and Diaconis, and Chung and Graham have used the representation theory of a class of monoids called LRBs to study random walks on combinatorial structures like hyperplane arrangements, matroids and graphs. Further impetus to study LRB algebras is given by the realization of Solomon's descent algebra for a finite Coxeter group W as the ring of W-invariants of the associated hyperplane LRB algebra. In this talk we discuss a surprising connection between the global dimension of LRB algebras and the Leray number of a simplicial complex. Leray numbers arise in both combinatorial topology and commutative algebra.