# A Garside presentation for Artin-Tits groups of type ${\tilde{C}}_{n}$

F. Digne^{[1]}

- [1] CNRS et Université de Picardie-Jules Verne LAMFA 33, Rue Saint-Leu 80039 Amiens Cedex (France)

Annales de l’institut Fourier (2012)

- Volume: 62, Issue: 2, page 641-666
- ISSN: 0373-0956

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topDigne, F.. "A Garside presentation for Artin-Tits groups of type $\widetilde{C}_n$." Annales de l’institut Fourier 62.2 (2012): 641-666. <http://eudml.org/doc/251119>.

@article{Digne2012,

abstract = {We prove that an Artin-Tits group of type $\widetilde\{C\}$ is the group of fractions of a Garside monoid, analogous to the known dual monoids associated with Artin-Tits groups of spherical type and obtained by the “generated group” method. This answers, in this particular case, a general question on Artin-Tits groups, gives a new presentation of an Artin-Tits group of type $\widetilde\{C\}$, and has consequences for the word problem, the computation of some centralizers or the triviality of the center. A key point of the proof is to show that this group is a group of fixed points in an Artin-Tits group of type $\widetilde\{A\}$ under an involution. Another important point is the study of the Hurwitz action of the usual braid group on the decomposition of a Coxeter element into a product of reflections.},

affiliation = {CNRS et Université de Picardie-Jules Verne LAMFA 33, Rue Saint-Leu 80039 Amiens Cedex (France)},

author = {Digne, F.},

journal = {Annales de l’institut Fourier},

keywords = {Braids; Garside; Artin-Tits groups; affine Coxeter groups; Garside monoids; Artin monoids; dual monoids; normal form theorems; centralizers; Coxeter elements; groups of quotients; braid groups; presentations; products of reflections},

language = {eng},

number = {2},

pages = {641-666},

publisher = {Association des Annales de l’institut Fourier},

title = {A Garside presentation for Artin-Tits groups of type $\widetilde\{C\}_n$},

url = {http://eudml.org/doc/251119},

volume = {62},

year = {2012},

}

TY - JOUR

AU - Digne, F.

TI - A Garside presentation for Artin-Tits groups of type $\widetilde{C}_n$

JO - Annales de l’institut Fourier

PY - 2012

PB - Association des Annales de l’institut Fourier

VL - 62

IS - 2

SP - 641

EP - 666

AB - We prove that an Artin-Tits group of type $\widetilde{C}$ is the group of fractions of a Garside monoid, analogous to the known dual monoids associated with Artin-Tits groups of spherical type and obtained by the “generated group” method. This answers, in this particular case, a general question on Artin-Tits groups, gives a new presentation of an Artin-Tits group of type $\widetilde{C}$, and has consequences for the word problem, the computation of some centralizers or the triviality of the center. A key point of the proof is to show that this group is a group of fixed points in an Artin-Tits group of type $\widetilde{A}$ under an involution. Another important point is the study of the Hurwitz action of the usual braid group on the decomposition of a Coxeter element into a product of reflections.

LA - eng

KW - Braids; Garside; Artin-Tits groups; affine Coxeter groups; Garside monoids; Artin monoids; dual monoids; normal form theorems; centralizers; Coxeter elements; groups of quotients; braid groups; presentations; products of reflections

UR - http://eudml.org/doc/251119

ER -

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