Garside families in Artin–Tits monoids and low elements in Coxeter groups

Patrick Dehornoy; Matthew Dyer; Christophe Hohlweg

doi:10.1016/j.crma.2015.01.008

Group theory

Garside families in Artin–Tits monoids and low elements in Coxeter groups
[Familles de Garside dans les monoïdes d'Artin–Tits et éléments bas d'un groupe de Coxeter]

Présenté par : the Editorial Board

Patrick Dehornoy ¹ ; Matthew Dyer ² ; Christophe Hohlweg ³

¹ Laboratoire de Mathématiques Nicolas-Oresme, CNRS UMR 6139, Université de Caen, 14032 Caen, France
² Department of Mathematics, University of Notre Dame, 255 Hurley, Notre Dame, IN 46556, USA
³ Université du Québec à Montréal, LaCIM et Département de mathématiques, CP 8888 Succ. Centre-ville, Montréal, Québec, H3C 3P8, Canada

Comptes Rendus. Mathématique, Volume 353 (2015) no. 5, pp. 403-408.

Résumé
Abstract

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 $(W, S)$ 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 $(W, S)$ with S finite includes S and is finite and closed under suffix and join with respect to the right weak order.

Reçu le : 2014-11-27
Accepté le : 2015-01-21
Publié le : 2015-03-18

DOI : 10.1016/j.crma.2015.01.008

Affiliations des auteurs :

Patrick Dehornoy ¹ ; Matthew Dyer ² ; Christophe Hohlweg ³

¹ Laboratoire de Mathématiques Nicolas-Oresme, CNRS UMR 6139, Université de Caen, 14032 Caen, France
² Department of Mathematics, University of Notre Dame, 255 Hurley, Notre Dame, IN 46556, USA
³ Université du Québec à Montréal, LaCIM et Département de mathématiques, CP 8888 Succ. Centre-ville, Montréal, Québec, H3C 3P8, Canada

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     title = {Garside families in {Artin{\textendash}Tits} monoids and low elements in {Coxeter} groups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {403--408},
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     doi = {10.1016/j.crma.2015.01.008},
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Patrick Dehornoy; Matthew Dyer; Christophe Hohlweg. Garside families in Artin–Tits monoids and low elements in Coxeter groups. Comptes Rendus. Mathématique, Volume 353 (2015) no. 5, pp. 403-408. doi : 10.1016/j.crma.2015.01.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.01.008/

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