Left regular bands with symmetry

Abstract

Left regular bands (or LRBs) are a well-studied class of finite semigroups which have connections with Markov chains, combinatorics, and discrete geometry. Although their semigroup algebras are rarely semisimple, much of their representation theoretic and homological data is well-understood. One tool which has been particularly useful in making these structures so tractable is poset topology, as developed in the wealth of work on LRBs put forth by Margolis--Saliola--Steinberg. Many LRBs in the literature carry symmetries, meaning they come equipped with groups that naturally act by semigroup automorphisms. In such a setting, it is natural to examine the invariant subalgebra of the LRB semigroup algebra as well as the invariant theory of the entire semigroup algebra. For some LRBs, interesting combinatorial structures---like Solomon's descent algebra and the derangement symmetric function---are known to arise when studying these topics. In this thesis, inspired by the work of Margolis--Saliola--Steinberg, we study LRBs with symmetry through the lens of group-equivariant poset topology. We first characterize when the invariant subalgebra of an LRB semigroup algebra is commutative or semisimple. Then, to understand the Cartan invariants of the invariant subalgebra as well as the invariant theory of the entire algebra, we study special decompositions of the semigroup algebras called invariant Peirce decompositions. For LRBs associated to hyperplane arrangements, there is a known connection between invariant Peirce decompositions and equivariant poset topology. In the case of the braid arrangement, we leverage this connection to carry out an in-depth analysis of the invariant Peirce decompositions. Our study both generalizes and unites work of Garsia--Reutenauer on the Cartan invariants of Solomon's descent algebra and Uyemura-Reyes on representations arising from shuffling processes. In joint work with B. Steinberg, we generalize the known connection between invariant Peirce decompositions and equivariant poset topology from LRBs associated to hyperplane arrangements to a larger class of LRBs known as CW LRBs. We then apply our generalization to understand the invariant Peirce decompositions for LRBs associated to CAT(0)-cube complexes, an exciting new example of CW LRBs recently discovered by Margolis--Saliola--Steinberg and Bandelt--Chepoi--Knauer. Additionally---and also in joint work with Steinberg---we develop a formula for the invariant Peirce decompositions of LRBs with hereditary semigroup algebras in terms of equivariant poset topology. We show that our formula recovers past work of the author with S. Brauner and V. Reiner on the free LRB and its q-analogue. Along the way, we find a new formula for the derangement symmetric function. Finally, an expository goal of this thesis is to explore examples of LRBs associated to CAT(0)-cube complexes. In particular, we explicitly work out the semigroup structure for LRBs arising from an especially combinatorial subclass of CAT(0)-cube complexes; this subclass includes Billera--Holmes--Vogtmann's space of phylogenetic trees.