Speaker: Patricia Commins (University of Minnesota)
Title: Invariant theory for the face algebra of the braid arrangement
Abstract: The hyperplanes of the braid arrangement subdivide n-dimensional Euclidean space into faces. These faces turn out to carry a monoid structure, and the associated monoid algebra -- the face algebra -- is well-studied, especially in relation to card shuffling and other Markov chains.
In this talk, we explore the action of the symmetric group on the face algebra. Bidigare proved the invariant subalgebra of the face algebra is (anti)isomorphic to a well-known subalgebra of the symmetric group algebra called Solomon's descent algebra. We answer the more general question: what is the structure of the face algebra as a simultaneous representation of the symmetric group and Solomon's descent algebra?