We will give a gentle introduction to a class of finite semigroups
bearing the cryptic name "left-regular bands" (LRBs). These LRBs show up, for example, in the combinatorics of reflection groups and hyperplane arrangements, in the analysis of mixing times for certain card-shuffling Markov chains, as well as in the space of phylogenetic trees.
We focus on examples with large groups of symmetries that act on the semigroup algebra of the LRB. Here the well-understood LRB representation theory, together with some combinatorics, allow one to answer two invariant-theory questions: What is the structure of the invariant subalgebra, and how does it act on the whole semigroup algebra?
This is based on joint work with Sarah Brauner and Patty Commins (arXiv:2206.11406).