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In recent years it has been noted that a number of combinatorial structuressuch as real and complex hyperplane arrangements, interval greedoids, matroidsand oriented matroids have the structure of a finite monoid called a leftregular band. Random walks on the monoid model a number of interesting Markovchains such as the Tsetlin library and riffle shuffle. The representationtheory of left regular bands then comes into play and has had a major influenceon both the combinatorics and the probability theory associated to suchstructures. In a recent paper, the authors established a close connectionbetween algebraic and combinatorial invariants of a left regular band byshowing that certain homological invariants of the algebra of a left regularband coincide with the cohomology of order complexes of posets naturallyassociated to the left regular band.  The purpose of the present monograph is to further develop and deepen theconnection between left regular bands and poset topology. This allows us tocompute finite projective resolutions of all simple modules of left regularband algebras over fields and much more. In the process, we are led to definethe class of CW left regular bands as the class of left regular bands whoseassociated posets are the face posets of regular CW complexes. Most of theexamples that have arisen in the literature belong to this class. A new andimportant class of examples is a left regular band structure on the face posetof a CAT(0) cube complex. Also, the recently introduced notion of a COM(complex of oriented matroids or conditional oriented matroid) fits nicely intoour setting and includes CAT(0) cube complexes and certain more general CAT(0)zonotopal complexes. A fairly complete picture of the representation theory forCW left regular bands is obtained.

Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry