Combinatorial topology and the global dimension of algebras arising in combinatorics

Margolis, Stuart, Saliola, Franco, and Steinberg, Benjamin. "Combinatorial topology and the global dimension of algebras arising in combinatorics." Journal of the European Mathematical Society 017.12 (2015): 3037-3080. <http://eudml.org/doc/277637>.

@article{Margolis2015,
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.},
author = {Margolis, Stuart, Saliola, Franco, Steinberg, Benjamin},
journal = {Journal of the European Mathematical Society},
keywords = {global dimension; hereditary algebra; cohomology; classifying space; left regular band; hyperplane arrangements; order complex; Leray number; chordal graph; global dimension; hereditary algebra; cohomology; classifying space; left regular band; hyperplane arrangements; order complex; Leray number; chordal graph},
language = {eng},
number = {12},
pages = {3037-3080},
publisher = {European Mathematical Society Publishing House},
title = {Combinatorial topology and the global dimension of algebras arising in combinatorics},
url = {http://eudml.org/doc/277637},
volume = {017},
year = {2015},
}

TY - JOUR
AU - Margolis, Stuart
AU - Saliola, Franco
AU - Steinberg, Benjamin
TI - Combinatorial topology and the global dimension of algebras arising in combinatorics
JO - Journal of the European Mathematical Society
PY - 2015
PB - European Mathematical Society Publishing House
VL - 017
IS - 12
SP - 3037
EP - 3080
AB - 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.
LA - eng
KW - global dimension; hereditary algebra; cohomology; classifying space; left regular band; hyperplane arrangements; order complex; Leray number; chordal graph; global dimension; hereditary algebra; cohomology; classifying space; left regular band; hyperplane arrangements; order complex; Leray number; chordal graph
UR - http://eudml.org/doc/277637
ER -