Explicit presentations for the dual braid monoids

Matthieu Picantin

doi:10.1016/S1631-073X(02)02370-1

Explicit presentations for the dual braid monoids
[Présentations pour les monoı̈des de tresses duaux]

Matthieu Picantin ¹

¹ SDAD, Département de mathématiques, Université de Caen, 14000 Caen, France

Comptes Rendus. Mathématique, Volume 334 (2002) no. 10, pp. 843-848.

Abstract (VO)
Résumé

Birman, Ko and Lee have introduced a new monoid $ℬ_{n}^{*}$ —with an explicit presentation—whose group of fractions is the n-strand braid group $ℬ_{n}$ . Building on a new approach by Digne, Michel and himself, Bessis has defined a dual braid monoid for every finite Coxeter type Artin–Tits group extending the type A case. Here, we give an explicit presentation for this dual braid monoid in the case of types B and D, and we study the combinatorics of the underlying Garside structures.

Birman, Ko et Lee ont introduit un nouveau monoı̈ de $ℬ_{n}^{*}$ —avec une présentation explicite—dont le groupe de fractions est le groupe $ℬ_{n}$ des tresses à n brins. Suivant une nouvelle approche proposée avec Digne et Michel, Bessis a défini un monoı̈ de de tresses dual pour tout groupe d'Artin–Tits de type de Coxeter fini généralisant le cas du type A. Ici, nous donnons une présentation explicite de ce monoı̈de de tresses dual pour les groupes d'Artin–Tits de type B et D, et nous étudions la combinatoire des structures de Garside sous-jacentes.

Accepté le : 2002-03-04
Publié le : 2002-04-12

DOI : 10.1016/S1631-073X(02)02370-1

Affiliations des auteurs :

Matthieu Picantin ¹

¹ SDAD, Département de mathématiques, Université de Caen, 14000 Caen, France

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Matthieu Picantin. Explicit presentations for the dual braid monoids. Comptes Rendus. Mathématique, Volume 334 (2002) no. 10, pp. 843-848. doi : 10.1016/S1631-073X(02)02370-1. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02370-1/

Bibliographie
Cité par

[1] D. Allcock, Braid pictures for Artin groups, ArXiv:math.GT/9907194, Trans. Amer. Math. Soc., to appear

[2] J. Birman; K.H. Ko; S.J. Lee A new approach to the word and conjugacy problems in the braid groups, Adv. Math., Volume 139 (1998), pp. 322-353

[3] D. Bessis, The dual braid monoid, ArXiv:math.GR/0101158

[4] D. Bessis, F. Digne, J. Michel, Springer theory in braid groups and the Birman–Ko–Lee monoid, ArXiv:math.GR/0010254

[5] T. Brady, C. Watt, K(π,1)'s for Artin groups of finite type, ArXiv:math.GR/0102078, Geom. Dedicata, to appear

[6] E. Brieskorn; K. Saito Artin-Gruppen und Coxeter-Gruppen, Invent. Math., Volume 17 (1972), pp. 245-271

[7] R. Charney, J. Meier, K. Whittlesey, Bestvina's normal form complex and the homology of Garside groups, ArXiv:math.GR/0202228

[8] R. Corran, Personal communication

[9] P. Dehornoy Braids and Self-Distributivity, Progress in Math., 192, Birkhäuser, 2000

[10] P. Dehornoy, Groupes de Garside, ArXiv:math.GR/0111157, Ann. Sci. École Norm. Sup., to appear

[11] P. Dehornoy, Complete positive group presentations, ArXiv:math.GR/0111275

[12] P. Dehornoy; L. Paris Gaussian groups and Garside groups, two generalizations of Artin groups, Proc. London Math. Soc., Volume 79 (1999) no. 3, pp. 569-604

[13] P. Dehornoy, Y. Lafont, Homology of Gaussian groups, ArXiv:math.GR/0111231

[14] P. Deligne Les immeubles des groupes de tresses généralisés, Invent. Math., Volume 17 (1972), pp. 273-302

[15] F.A. Garside The braid group and other groups, Quart. J. Math. Oxford, Volume 20 (1969), pp. 235-254

[16] M. Picantin The conjugacy problem in small Gaussian groups, Comm. Algebra, Volume 29 (2001) no. 3, pp. 1021-1039

[17] M. Picantin The center of thin Gaussian groups, J. Algebra, Volume 245 (2001) no. 1, pp. 92-122

[18] M. Picantin, Petits groupes gaussiens, Ph.D. thesis, Université de Caen, 2000

[19] V. Reiner Non-crossing partitions for classical reflection groups, Disc. Math., Volume 177 (1997), pp. 195-222

[20] The GAP Group, GAP – Groups, Algorithms, and Programming, V.4.2, 2000, http://www.gap-system.org

S. Abbes; S. Gouëzel; V. Jugé; J. Mairesse Asymptotic combinatorics of Artin–Tits monoids and of some other monoids, Journal of Algebra, Volume 525 (2019), p. 497 | DOI:10.1016/j.jalgebra.2019.01.019
Oussama Ajbal Fixed Points of Endomorphisms and Metrics in preGarside Monoids, Communications in Algebra, Volume 44 (2016) no. 6, p. 2685 | DOI:10.1080/00927872.2015.1065842
Vivien Ripoll Lyashko–Looijenga morphisms and submaximal factorizations of a Coxeter element, Journal of Algebraic Combinatorics, Volume 36 (2012) no. 4, p. 649 | DOI:10.1007/s10801-012-0354-4
Ping Zhu; Qiaoyan Wen Affine braid groups: a better platform than braid groups for cryptology?, Applicable Algebra in Engineering, Communication and Computing, Volume 22 (2011) no. 5-6, p. 375 | DOI:10.1007/s00200-011-0157-1
Nathan Reading Clusters, Coxeter-sortable elements and noncrossing partitions, Transactions of the American Mathematical Society, Volume 359 (2007) no. 12, p. 5931 | DOI:10.1090/s0002-9947-07-04319-x
Christos A. Athanasiadis; Victor Reiner Noncrossing Partitions for the GroupDn, SIAM Journal on Discrete Mathematics, Volume 18 (2004) no. 2, p. 397 | DOI:10.1137/s0895480103432192
Patrick Dehornoy Disks in trivial braid diagrams, Topology, Volume 43 (2004) no. 5, p. 1067 | DOI:10.1016/j.top.2003.11.005
Patrick Dehornoy Complete positive group presentations, Journal of Algebra, Volume 268 (2003) no. 1, p. 156 | DOI:10.1016/s0021-8693(03)00326-0
Matthieu Picantin Automatic Structures for Torus Link Groups, Journal of Knot Theory and Its Ramifications, Volume 12 (2003) no. 06, p. 833 | DOI:10.1142/s0218216503002627

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