This talk is a summary of work done over the past decade by the speaker, Franco Saliola and Benjamin Steinberg.

In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. This brings the representation theory of left regular band algebras into play. The authors proved that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band.

The purpose of the talk is to discuss the deep connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all simple modules of unital left regular band algebras over fields, and much more. We define the class of CW left regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the examples that have arisen in the literature are in this class. A new and important class of examples is a left regular band structure on the face post of CAT(0) cube complexes and generalizations. A fairly complete picture of the representation theory for CW left regular bands is given.