JournalscmhVol. 81 , No. 1DOI 10.4171/cmh/41

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

  • François Digne

    Université de Picardie Jules-Verne, Amiens, France
Présentations duales des groupes de tresses de type affine <em>Ã</em> cover

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\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 WW is a Coxeter group, one can consider the length lRl_R on WW with respect to the generating set RR consisting of all reflections. Let cc be a Coxeter element in WW and let PcP_c be the set of elements pWp\in W such that cc can be written c=ppc=pp' with lR(c)=lR(p)+lR(p)l_R(c)=l_R(p)+l_R(p'). We define the monoid M(Pc)M(P_c) to be the monoid generated by a set Pc\underline P_c in one-to-one correspondence, ppp\mapsto \underline p, with PcP_c with only relations pp=p.p\underline{pp'}=\underline p.\underline p' whenever pp, pp' and pppp' are in PcP_c and lR(pp)=lR(p)+lR(p)l_R(pp')=l_R(p)+l_R(p'). We conjecture that the group of quotients of M(Pc)M(P_c) is the Artin--Tits group associated to WW 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~\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 AA.