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On the generic Krammer representations. (Sur les représentations de Krammer génériques.) (French) Zbl 1183.20036
Summary: We define a representation of the Artin groups of type \(ADE\) by monodromy of generalized KZ-systems which is shown to be isomorphic to the generalized Krammer representation originally defined by A. M. Cohen and D. Wales, and independently by F. Digne. It follows that all pure Artin groups of spherical type are residually torsion-free nilpotent, hence (bi-)orderable. Using that construction we show that these irreducible representations are Zariski-dense in the corresponding general linear group. It follows that all irreducible Artin groups of spherical type can be embedded as Zariski-dense subgroups of some general linear group. As group-theoretical applications we prove properties of non-decomposition in direct products for several subgroups of Artin groups, and a generalization in arbitrary types of a celebrated property of D. Long for the braid groups. We also determine the Frattini and Fitting subgroups and discuss unitarity properties of the representations.

MSC:
20F36 Braid groups; Artin groups
20C15 Ordinary representations and characters
20E07 Subgroup theorems; subgroup growth
20E26 Residual properties and generalizations; residually finite groups
32G34 Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation)
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