2004/02/07 by François Digne, Digne, François
Mathematics · #20F36 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Geometric Topology (math.GT) #Group Theory (math.GR) #math.GR #math.GT #msc:20F36
paper · pdf · doi:10.48550/arxiv.math/0402116
openalex publication_date 2004/02/07 · arxiv created 2005/09/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the present paper we define dual monoids for all Artin-Tits groups and we prove that for the type An we get a (quasi)-Garside structure. Such a structure provides normal forms for the Artin-Tits group elements and allows to solve some questions such as to determine the centralizer of a power of the Coxeter element in the Artin-Tits group. More precisely, if W is a Coxeter group, one can consider the length lR on W with respect to the generating set R consisting of all reflections. Let c be a Coxeter element in W and let Pc be the set of elements p∈ W such that c can be written c=pp' with lR(c)=lR(p)+lR(p'). We define the monoid M(Pc) to be the monoid generated by a set \underline Pc in one-to-one correspondence, p↦ \underline p, with Pc with only relations \underlinepp'=\underline p.\underline p' whenever p, p' and pp' are in Pc and lR(pp')=lR(p)+lR(p'). We conjecture that the group of quotients of M(Pc) is the Artin-Tits group associated to W and that it has a simple presentation (see \refconjecture (ii)). These conjectures are known to be true for spherical type Artin-Tits groups. Here we prove them for Artin-Tits groups of type A. Moreover, we show that for exactly one choice of the Coxeter element (up to diagram automorphism) we obtain a (quasi-) Garside monoid. The proof makes use of non-crossing paths in an annulus which are the counterpart in this context of the non-crossing partitions used for type A.