Root systems, symmetries and linear representations of Artin groups
Annales mathématiques Blaise Pascal, Volume 26 (2019) no. 1, pp. 25-54.

Let Γ be a Coxeter graph, let W be its associated Coxeter group, and let G be a group of symmetries of Γ. Recall that, by a theorem of Hée and Mühlherr, W G is a Coxeter group associated to some Coxeter graph Γ ^. We denote by Φ + the set of positive roots of Γ and by Φ ^ + the set of positive roots of Γ ^. Let E be a vector space over a field 𝕂 having a basis in one-to-one correspondence with Φ + . The action of G on Γ induces an action of G on Φ + , and therefore on E. We show that E G contains a linearly independent family of vectors naturally in one-to-one correspondence with Φ ^ + and we determine exactly when this family is a basis of E G . This question is motivated by the construction of Krammer’s style linear representations for non simply laced Artin groups.

Soient Γ un graphe de Coxeter, W son groupe de Coxeter associé et G un groupe de symétries de Γ. Rappelons que, par un théorème de Hée et Mühlherr, W G est un groupe de Coxeter associé à un certain graphe de Coxeter Γ ^. On note Φ + l’ensemble des racines positives de Γ et Φ ^ + l’ensemble des racines positives de Γ ^. Soit E un espace vectoriel sur un corps 𝕂 ayant une base en bijection avec Φ + . L’action de G sur Γ induit une action de G sur Φ + , et donc sur E. Nous montrons que E G contient une famille libre de vecteurs naturellement en bijection avec Φ ^ + et nous déterminons exactement quand cette famille est une base de E G . Cette question est motivée par la construction de représentations linéaires à la Krammer de groupes d’Artin non simplement lacés.

Published online:
DOI: 10.5802/ambp.381
Classification: 20F36
Keywords: Artin group, linear representation, Coxeter group, root system
Olivier Geneste 1; Jean-Yves Hée 2; Luis Paris 1

1 IMB, UMR 5584, CNRS Université Bourgogne Franche-Comté 21000 Dijon FRANCE
2 LAMFA, UMR 7352, CNRS Université de Picardie Jules Verne 80039 Amiens FRANCE
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Olivier Geneste; Jean-Yves Hée; Luis Paris. Root systems, symmetries and linear representations of Artin groups. Annales mathématiques Blaise Pascal, Volume 26 (2019) no. 1, pp. 25-54. doi : 10.5802/ambp.381. https://ambp.centre-mersenne.org/articles/10.5802/ambp.381/

[1] Stephen J. Bigelow Braid groups are linear, J. Am. Math. Soc., Volume 14 (2001) no. 2, pp. 471-486 | DOI | MR | Zbl

[2] Nicolas Bourbaki Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, 1337, Hermann, 1968 | Zbl

[3] Anatole Castella Automorphismes et admissibilité dans les groupes de Coxeter et les monoïdes d’Artin-Tits, Université de Paris-Sud (France) (2006) (Ph. D. Thesis) | Zbl

[4] Anatole Castella On Lawrence-Krammer representations, J. Algebra, Volume 322 (2009) no. 10, pp. 3614-3639 | DOI | MR | Zbl

[5] Anatole Castella Twisted Lawrence-Krammer representations (2017) (https://arxiv.org/abs/1711.09860) | Zbl

[6] Arjeh M. Cohen; David B. Wales Linearity of Artin groups of finite type, Isr. J. Math., Volume 131 (2002), pp. 101-123 | DOI | MR | Zbl

[7] John Crisp Symmetrical subgroups of Artin groups, Adv. Math., Volume 152 (2000) no. 1, pp. 159-177 | DOI | MR | Zbl

[8] John Crisp Erratum to: “Symmetrical subgroups of Artin groups”, Adv. Math., Volume 179 (2003) no. 2, pp. 318-320 | DOI | MR | Zbl

[9] Michael W. Davis The geometry and topology of Coxeter groups, London Mathematical Society Monographs, 32, Princeton University Press, 2008 | MR | Zbl

[10] Patrick Dehornoy; Luis Paris Gaussian groups and Garside groups, two generalisations of Artin groups, Proc. Lond. Math. Soc., Volume 79 (1999) no. 3, pp. 569-604 | DOI | MR

[11] Vinay V. Deodhar On the root system of a Coxeter group, Commun. Algebra, Volume 10 (1982) no. 6, pp. 611-630 | DOI | MR | Zbl

[12] François Digne On the linearity of Artin braid groups, J. Algebra, Volume 268 (2003) no. 1, pp. 39-57 | DOI | MR | Zbl

[13] Olivier Geneste; Luis Paris Coxeter groups, symmetries, and rooted representations, Commun. Algebra, Volume 46 (2018) no. 5, pp. 1996-2002 | DOI | MR | Zbl

[14] Jean-Yves Hée Système de racines sur un anneau commutatif totalement ordonné, Geom. Dedicata, Volume 37 (1991) no. 1, pp. 65-102 | Zbl

[15] Jean-Yves Hée Sur la torsion de Steinberg-Ree des groupes de Chevalley et des groupes de Kac-Moody, Thèse de Doctorat d’État, Université de Paris-Sud (France), 1993

[16] Jean-Yves Hée Une démonstration simple de la fidélité de la représentation de Lawrence-Krammer-Paris, J. Algebra, Volume 321 (2009) no. 3, pp. 1039-1048 | Zbl

[17] James E. Humphreys Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, 29, Cambridge University Press, 1990 | MR | Zbl

[18] Victor G. Kac Infinite-dimensional Lie algebras, Cambridge University Press, 1990 | Zbl

[19] Daan Krammer The conjugacy problem for Coxeter groups, Universiteit Utrecht (Netherlands) (1994) (Ph. D. Thesis) | Zbl

[20] Daan Krammer Braid groups are linear, Ann. Math., Volume 155 (2002) no. 1, pp. 131-156 | DOI | MR | Zbl

[21] Daan Krammer The conjugacy problem for Coxeter groups, Groups Geom. Dyn., Volume 3 (2009) no. 1, pp. 71-171 | DOI | MR | Zbl

[22] Ruth J. Lawrence Homological representations of the Hecke algebra, Commun. Math. Phys., Volume 135 (1990) no. 1, pp. 141-191 | DOI | MR | Zbl

[23] Jean Michel A note on words in braid monoids, J. Algebra, Volume 215 (1999) no. 1, pp. 366-377 | DOI | MR | Zbl

[24] B. Mühlherr Coxeter groups in Coxeter groups, Finite geometry and combinatorics (Deinze, 1992) (London Mathematical Society Lecture Note Series), Volume 191, Cambridge University Press, 1993, pp. 277-287 | DOI | MR | Zbl

[25] Luis Paris Artin monoids inject in their groups, Comment. Math. Helv., Volume 77 (2002) no. 3, pp. 609-637 | DOI | MR | Zbl

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