Présentations duales des groupes de tresses de type affine $\tilde A$
| dc.creator | Digne, François | |
| dc.date | 2004-02-07 | |
| dc.date | 2005-09-06 | |
| dc.date.accessioned | 2026-07-07T05:05:14Z | |
| dc.date.available | 2026-07-07T05:05:14Z | |
| dc.description | 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 $c=pp'$ 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 $\underline P_c$ in one-to-one correspondence, $p\mapsto \underline p$, with $P_c$ with only relations $\underline{pp'}=\underline p.\underline p'$ 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 \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 $A$. | |
| dc.identifier | https://arxiv.org/abs/math/0402116 | |
| dc.identifier | http://arxiv.org/abs/math/0402116 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/70092 | |
| dc.subject | Group Theory | |
| dc.subject | Geometric Topology | |
| dc.subject | 20F36 | |
| dc.title | Présentations duales des groupes de tresses de type affine $\tilde A$ | |
| dc.type | text |