Combinatorial Topology and the Global Dimension of Algebras Arising in Combinatorics

TL;DR

This paper links algebraic properties of left regular band algebras with combinatorial invariants of associated posets and complexes, revealing bounds on global dimension through topological invariants.

Contribution

It establishes a connection between the homological invariants of left regular band algebras and the cohomology of associated order complexes, providing bounds on global dimension.

Findings

Global dimension bounded by Leray number of order complex

Homological invariants coincide with cohomology of poset complexes

Constructs a left regular band with algebra's global dimension equal to Leray number

Abstract

In a highly influential paper, Bidigare, Hanlon and Rockmore showed that a number of popular Markov chains are random walks on the faces of a hyperplane arrangement. Their analysis of these Markov chains took advantage of the monoid structure on the set of faces. This theory was later extended by Brown to a larger class of monoids called left regular bands. In both cases, the representation theory of these monoids played a prominent role. In particular, it was used to compute the spectrum of the transition operators of the Markov chains and to prove diagonalizability of the transition operators. In this paper, we establish a close connection between algebraic and combinatorial invariants of a left regular band: we show 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. For instance, we show that the global dimension of these algebras is bounded above by the Leray number of the associated order complex. Conversely, we associate to every flag complex a left regular band whose algebra has global dimension precisely the Leray number of the flag complex.