Left regular bands (LRBs) are a special family of finite semigroups which appear surprisingly frequently in algebraic combinatorics. Originally studied for their connections to Markov chains, LRBs have more recently been studied for the combinatorial representation theory of their semigroup algebras. Many of the LRBs in the literature come equipped with natural groups acting by semigroup automorphisms. In such cases, one can ask

• (i) What is the structure of the invariant subalgebra of the LRB semigroup algebra? and
• (ii) what is the structure of the LRB semigroup algebra as a simultaneous representation of its invariant subalgebra and the group?

In this talk, we will discuss joint work with Benjamin Steinberg which approaches these questions with group-equivariant poset topology. This generalizes recent work of the speaker on the face monoid of the braid arrangement and recent work of the speaker, Sarah Brauner, and Vic Reiner on the free left regular band and its q-analog.