Date: 04/08/2022

Speaker: Patricia Commins

Invariant Theory of the Free Left Regular Band
Bidigare, Hanlon, and Rockmore developed a unified theory for understanding the eigenvalues of a family of shuffling operators connected to hyperplane arrangements. We begin this talk by introducing their work and Brown's generalization to a class of semigroups known as "left regular bands." We will then focus on the "free left regular band," a semigroup formed from words without repeated letters. In particular, we will examine the action of the symmetric group $S_n$ on its semigroup algebra, simultaneously decomposing the semigroup algebra as a module over $S_n$ and its $S_n$-invariant subalgebra. Many nice combinatorial objects such as the lattices, derangements, and Stirling numbers will appear along the way! This is joint work with Sarah Brauner and Vic Reiner.