Abstract
When Daan Krammer and Stephen Bigelow independently proved that braid groups are linear, they used the Lawrence–Krammer–Bigelow representation for generic values of its variables q and t. The t variable is closely connected to the traditional Garside structure of the braid group and plays a major role in Krammer’s algebraic proof. The q variable, associated with the dual Garside structure of the braid group, has received less attention. In this article we give a geometric interpretation of the q portion of the LKB representation in terms of an action of the braid group on the space of non-degenerate euclidean simplices. In our interpretation, braid group elements act by systematically reshaping (and relabeling) euclidean simplices. The reshapings associated to the simple elements in the dual Garside structure of the braid group are of an especially elementary type that we call relabeling and rescaling.
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References
D. Armstrong, Generalized noncrossing partitions and combinatorics of Coxeter groups. Mem. Am. Math. Soc. 202 (949), x+159 (2009)
S. Bigelow, The Burau representation is not faithful for n = 5. Geom. Topol. 3, 397–404 (1999)
S.J. Bigelow, Braid groups are linear. J. Am. Math. Soc. 14 (2), 471–486 (2001)
S. Bigelow, The Lawrence–Krammer representation, in Topology and Geometry of Manifolds (Athens, GA, 2001). Proceedings of Symposia in Pure Mathematics, vol. 71 (American Mathematical Society, Providence, RI, 2003), pp. 51–68.
J. Birman, K.H. Ko, S.J. Lee, A new approach to the word and conjugacy problems in the braid groups. Adv. Math. 139 (2), 322–353 (1998)
N. Brady, J. McCammond, Factoring Euclidean isometries. Int. J. Algebra Comput. 25 (1–2), 325–347 (2015). doi:10.1142/S0218196715400135
A.M. Cohen, D.B. Wales, Linearity of Artin groups of finite type. Isr. J. Math. 131, 101–123 (2002)
F. Digne, On the linearity of Artin braid groups. J. Algebra 268 (1), 39–57 (2003)
T. Ito, B. Wiest, Lawrence–Krammer–Bigelow representations and dual Garside length of braids. Geom. Topol. 19 (3), 1361–1381 (2015). doi:10.2140/gt.2015.19.1361
D. Krammer, The braid group \(B_{4}\) is linear. Invent. Math. 142 (3), 451–486 (2000)
D. Krammer, Braid groups are linear. Ann. Math. (2) 155 (1), 131–156 (2002)
D. Margalit, J. McCammond, Geometric presentations for the pure braid group. J. Knot Theor. Ramif. 18 (1), 1–20 (2009)
J. McCammond, Noncrossing partitions in surprising locations. Am. Math. Mon. 113 (7), 598–610 (2006)
L. Paris, Artin monoids inject in their groups. Comment. Math. Helv. 77 (3), 609–637 (2002)
R. Simion, D. Ullman, On the structure of the lattice of noncrossing partitions. Discrete Math. 98 (3), 193–206 (1991)
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Chisholm, E.L., McCammond, J. (2016). Braid Groups and Euclidean Simplices. In: Callegaro, F., Cohen, F., De Concini, C., Feichtner, E., Gaiffi, G., Salvetti, M. (eds) Configuration Spaces. Springer INdAM Series, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-319-31580-5_12
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DOI: https://doi.org/10.1007/978-3-319-31580-5_12
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