Discrete Geometry and Combinatorics Seminar
This is an expository talk presenting a beautiful interplay between combinatorial topology and homological algebra. The starting point is a class of monoids called "left regular bands" that arise naturally in algebraic combinatorics and that encode several notable Markov chains. The representation theory of monoids plays a prominent role in the analysis of these chains; for example, to compute the spectrum of the transition operators of the Markov chains and to prove the diagonalizability of the transition operators (via work by L. Billera, K. Brown and P. Diaconis). Further developments have uncovered a close connection between certain algebraic and combinatorial invariants of these monoids. More precisely, certain homological invariants of the monoid algebras ("Ext-spaces") coincide with the cohomology of order complexes of posets naturally associated with the monoids. We will present this interplay and explore some consequences. This talk is based on joint work with Stuart Margolis (Bar Ilan) and Benjamin Steinberg (CUNY).