Présentations duales des groupes de tresses de type affine $\tilde{A}$

Abstract

In the present paper we define dual monoids for all Artin-Tits groups and we prove that for the type ${\tilde{A}}_n$ 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 $p\in W$ such that $c$ can be written $c=p{p}^{\prime }$ with $l_R(c)=l_R(p)+l_R(p^{\prime })$. We define the monoid $M(P_c)$ to be the monoid generated by a set ${\underline{P}}_c$ in one-to-one correspondence, $p\mapsto \underline{p}$, with $P_c$ with only relations $\underline{p{p}^{\prime }}=\underline{p}.{\underline{p}}^{\prime }$ whenever $p$, $p^{\prime }$ and $p{p}^{\prime }$ are in $P_c$ and $l_R(p{p}^{\prime })=l_R(p)+l_R(p^{\prime })$. 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 \ref{conjecture} (ii)). These conjectures are known to be true for spherical type Artin-Tits groups. Here we prove them for Artin-Tits groups of type $\tilde{A}$. 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 $A$.