[Familles de Garside dans les monoïdes d'Artin–Tits et éléments bas d'un groupe de Coxeter]
Nous montrons que tout groupe d'Artin–Tits finiment engendré possède une famille de Garside finie, en introduisant la notion d'élément bas dans un groupe de Coxeter et en prouvant que, si est un système de Coxeter avec S fini, l'ensemble des éléments bas de W inclut S et est fini et clos par suffixe et borne supérieure dans l'ordre faible à droite.
We show that every finitely generated Artin–Tits group admits a finite Garside family, by introducing the notion of a low element in a Coxeter group and proving that the family of all low elements in a Coxeter system with S finite includes S and is finite and closed under suffix and join with respect to the right weak order.
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@article{CRMATH_2015__353_5_403_0,
author = {Dehornoy, Patrick and Dyer, Matthew and Hohlweg, Christophe},
title = {Garside families in {Artin{\textendash}Tits} monoids and low elements in {Coxeter} groups},
journal = {Comptes Rendus. Math\'ematique},
pages = {403--408},
publisher = {Elsevier},
volume = {353},
number = {5},
year = {2015},
doi = {10.1016/j.crma.2015.01.008},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2015.01.008/}
}
TY - JOUR AU - Dehornoy, Patrick AU - Dyer, Matthew AU - Hohlweg, Christophe TI - Garside families in Artin–Tits monoids and low elements in Coxeter groups JO - Comptes Rendus. Mathématique PY - 2015 SP - 403 EP - 408 VL - 353 IS - 5 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2015.01.008/ DO - 10.1016/j.crma.2015.01.008 LA - en ID - CRMATH_2015__353_5_403_0 ER -
%0 Journal Article %A Dehornoy, Patrick %A Dyer, Matthew %A Hohlweg, Christophe %T Garside families in Artin–Tits monoids and low elements in Coxeter groups %J Comptes Rendus. Mathématique %D 2015 %P 403-408 %V 353 %N 5 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2015.01.008/ %R 10.1016/j.crma.2015.01.008 %G en %F CRMATH_2015__353_5_403_0
Dehornoy, Patrick; Dyer, Matthew; Hohlweg, Christophe. Garside families in Artin–Tits monoids and low elements in Coxeter groups. Comptes Rendus. Mathématique, Tome 353 (2015) no. 5, pp. 403-408. doi : 10.1016/j.crma.2015.01.008. http://www.numdam.org/articles/10.1016/j.crma.2015.01.008/
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