2005/04/07 by Iván Marín, Ivan Marin, Marin, Ivan
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #math.GR #math.RT
paper · pdf · doi:10.48550/arxiv.math/0504143
29 pages, French ; full revision including many improvements
openalex publication_date 2005/04/07 · arxiv created 2006/06/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 independantly 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.