Left regular bands (LRBs) are a special class of finite semigroups. They are often studied for their connections to Markov chains, but have many interesting properties in their own right, such as a rich connection to poset topology developed by Margolis, Saliola, and Steinberg. Many of the LRBs appearing in the literature are naturally equipped with group actions. In this talk, we will explore LRB semigroup algebras under these group actions, focusing on the structure of the invariant subalgebra and more generally, the semigroup algebra as a simultaneous representation of the group and the invariant subalgebra.
Our primary example will be the face monoid of the braid arrangement which is acted upon by the symmetric group. Bidigare proved the invariant subalgebra of the face semigroup algebra is (anti-)isomorphic to Solomon’s descent algebra. We will explore the structure of the entire semigroup algebra as a simultaneous representation of the descent algebra and the symmetric group. If time permits, we may explore how this example extends to a more general class of LRBs.
No prior knowledge of LRBs or hyperplane arrangements will be assumed!
This video is part of the New York Group Theory Cooperative‘s group theory seminar series.
