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AN INTRODUCTION TO GARSIDE STRUCTURES JON MCCAMMOND 1
 

Summary: AN INTRODUCTION TO GARSIDE STRUCTURES
JON MCCAMMOND 1
Abstract. Geometric combinatorialists often study partially ordered sets in
which each covering relation has been assigned some sort of label. In this ar-
ticle we discuss how each such labeled poset naturally has a monoid, a group,
and a cell complex associated with it. Moreover, when the labeled poset sat-
isfies three simple combinatorial conditions, the connections among the poset,
monoid, group, and complex are particularly close and interesting. Posets
satisfying these three conditions are (roughly) equivalent to the notion of a
Garside structure for a group as developed recently within geometric group
theory by Patrick Dehornoy in [15]. The goal of this article is to provide a
quick introduction to the combinatorial version of this notion of a Garside
structure and, more specifically, to the particular combinatorial Garside struc-
tures which arise in the study of Coxeter groups and Artin groups. These are
the labeled partially ordered sets that combinatorialists know as the general-
ized non-crossing partition lattices.
Over the past several years, geometric group theorists have developed a theory
of Garside structures to help them better understand Artin's braid groups and their
generalizations. See for example [1, 2, 3, 7, 9, 14, 15, 16, 17]. Groups with Garside
structures are now emerging as a well-behaved class of groups worthy of study in

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics