# Présentations duales des groupes de tresses de type affine <em>Ã</em>

### François Digne

Université de Picardie Jules-Verne, Amiens, France

## Abstract

Artin--Tits groups of spherical type have two well-known Garside structures, coming respectively from the divisibility properties of the classical Artin monoid and of the dual monoid. For general Artin--Tits groups, the classical monoids have no such Garside property. In the present paper we define dual monoids for all Artin--Tits groups and we prove that for the type $A~_{n}$ 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 $l_{R}$ on $W$ with respect to the generating set $R$ consisting of all reflections. Let $c$ be a Coxeter element in $W$ and let $P_{c}$ be the set of elements $p∈W$ such that $c$ can be written $c=pp_{′}$ with $l_{R}(c)=l_{R}(p)+l_{R}(p_{′})$. We define the monoid $M(P_{c})$ to be the monoid generated by a set $P _{c}$ in one-to-one correspondence, $p↦p $, with $P_{c}$ with only relations $pp_{′} =p .p _{′}$ whenever $p$, $p_{′}$ and $pp_{′}$ are in $P_{c}$ and $l_{R}(pp_{′})=l_{R}(p)+l_{R}(p_{′})$. We conjecture that the group of quotients of $M(P_{c})$ is the Artin--Tits group associated to $W$ and that it has a simple presentation (see 1.1 (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 $A$.

## Cite this article

François Digne, Présentations duales des groupes de tresses de type affine <em>Ã</em>. Comment. Math. Helv. 81 (2006), no. 1, pp. 23–47

DOI 10.4171/CMH/41