Over the last 15 years, it has been noted that many combinatorial structures, such as real and complex hyperplane arrangements, interval greedoids, matroids, oriented matroids, and others have the structure of a left regular band. The representation theory of the associated band has had a major influence on understanding these objects along with related structures such as finite Coxeter groups and various Markov processes.
In return, this has spurred a deeper development of the representation theory and cohomology theory of left regular bands and more general classes of finite monoids. In particular, the Ext modules between simple LRB modules over a field turn out to be intimately related to the cohomology of the order complex of the R-order of the LRB and other related simplicial complexes. A more general class of examples than those mentioned above is the class of COMs recently defined by Bandelt, Chepoi and Knauer. These fit into the wider class of LRBs all of whose retractions (certain intervals in the poset) are isomorphic to face posets of regular CW complexes.
For this class of LRBs, we can compute a quiver presentation, the global dimension of the algebra and have an analogue of the Zaslavsky Theorem on counting faces of hyperplane arrangements. An intriguing more general example will be served up for a snack: the Hegelian taco as defined by Lawvere. Finally, a surprising connection to Leray numbers and partially commutative LRBs will be discussed.
N.B.: All terms will be defined in the talk
[Joint work with Franco Saliola (Université de Montreal) & Benjamin Steinberg (City College of New York)]