Hyperplane arrangement face algebras and their associated Markov chains.

Abstract

Let ${\cal A}$ be a hyperplane arrangement and let F and G be two of its faces. We define the product of F and G to be the smallest face whose closure contains F and which is separated from G by the fewest number of hyperplanes. Extending this product to linear combinations of faces gives us the face algebra of the arrangement ${\cal A}$. This thesis first analyzes the structures of these face algebras. We identify the nilradical for the face algebra of a general hyperplane arrangement and determine its nilpotency index. The action of the faces of a hyperplane arrangement on its chambers gives us a faithful representation of its face algebra. We develop an elegant formula for the eigenvalues of these chamber representations. The hyperplanes fixed by the reflections in a finite reflection group form a hyperplane arrangement called a reflection arrangement. There is a natural bijection between faces of a reflection arrangement and the left cosets of the parabolic sub-groups of its finite reflection group. We derive an algebraic formula for the product of two parabolic cosets. The elements of a reflection arrangement face algebra which are fixed by the finite reflection group form a subalgebra which is antiisomorphic to Solomon's descent algebra for the finite reflection group. Combinatorial descriptions for faces, edges and chambers are given for the two families of reflection arrangements corresponding to the symmetric and hyperoctahedral groups and the dimensions of their face algebras are determined. We also develop a formula for the eigenvalues of Solomon's symmetric descent algebra acting by right multiplication on the symmetric group algebra. The chamber representation gives us a class of Markov chains on the chambers of hyperplane arrangements. The natural correspondence we develop between the chambers of the symmetric group reflection arrangement and permutations allows us to study several Markov chains on permutations of objects. Our spectral results are used to determine the eigenvalues for several well-known Markov chains including riffle shuffling and the Tsetlin Library, as well as generate some new results.