The Lawrence–Krammer representation introduced by Lawrence and Krammer in order to show the linearity of the braid group is generically irreducible. We show this fact and show further that for some values of its two parameters, when these are specialized to complex numbers, the representation becomes reducible. We describe what these values are and give a complete description of the dimensions of the invariant subspaces when the representation is reducible.
La représentation de Lawrence–Krammer, introduite par Lawrence et Krammer pour montrer la linéarité du groupe de tresses, est génériquement irréductible. On montre ce fait et on montre également que lorsque les deux paramètres de la représentation prennent certaines valeurs complexes, la représentation devient réductible. On donne ici toutes les valeurs des paramètres pour lesquelles la représentation est réductible, ainsi que les dimensions des sous-espaces stables.
Accepted:
Published online:
Claire Levaillant 1
@article{CRMATH_2009__347_1-2_15_0, author = {Claire Levaillant}, title = {Irreducibility of the {Lawrence{\textendash}Krammer} representation of the {BMW} algebra of type $ {A}_{n-1}$}, journal = {Comptes Rendus. Math\'ematique}, pages = {15--20}, publisher = {Elsevier}, volume = {347}, number = {1-2}, year = {2009}, doi = {10.1016/j.crma.2008.11.011}, language = {en}, }
Claire Levaillant. Irreducibility of the Lawrence–Krammer representation of the BMW algebra of type $ {A}_{n-1}$. Comptes Rendus. Mathématique, Volume 347 (2009) no. 1-2, pp. 15-20. doi : 10.1016/j.crma.2008.11.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.11.011/
[1] Braid groups are linear, J. Amer. Math. Soc., Volume 14 (2001), pp. 471-486
[2] The Lawrence–Krammer representation, 2002 | arXiv
[3] BMW algebras of simply laced type, J. Algebra, Volume 285 (2005) no. 2, pp. 439-450
[4] Linearity of Artin groups of finite type, Israel J. Math., Volume 131 (2002), pp. 101-123
[5] On the minimal dimensions of irreducible representations of symmetric groups, Math. Proc. Camb. Phil. Soc., Volume 94 (1983), pp. 417-424
[6] Braid groups are linear, Ann. of Math., Volume 155 (2002), pp. 131-156
[7] C. Levaillant, Irreducibility of the BMW algebra of type , PhD thesis, California Institute of Technology, 2008, http://etd.caltech.edu/etd/available/etd-05292008-110016/
[8] A. Mathas, Iwahori–Hecke algebras and Schur algebras of the symmetric group, University Lecture Series, vol. 15
[9] I. Marin, Représentation linéaires des tresses infinitésimales, thèse de l'université d'Orsay, 2001
[10] Sur les représentations de Krammer génériques, Ann. Inst. Fourier, Volume 57 (2007) no. 6, pp. 1883-1925
[11] Quantum groups and subfactors of type B, C, and D, Commun. Math. Phys., Volume 133 (1990), pp. 383-432
[12] On Krammer's representation of the braid group, Math. Ann., Volume 321 (2001), pp. 197-211
Cited by Sources:
☆ This work is part of the author's PhD thesis at Caltech under the direction of Professor David B. Wales.
Comments - Policy