A Garside presentation for Artin-Tits groups of type 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

Abstract

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We prove that an Artin-Tits group of type 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 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 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.

How to cite

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Digne, 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 -

References

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