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Ivan Marin

Publications and source records attributed to Ivan Marin.

50 records · Page 3Linked to original sources
arXiv2008

Krammer representations for complex braid groups

Let B be the generalized braid group associated to some finite complex reflection group. We define a representation of B of dimension the number of reflections of the corresponding reflection group, which generalizes the Krammer representation of the classical braid groups, and is thus a good candidate in view of proving the linearity of these groups. We decompose this representation in irreducible components and compute its Zariski closure, as well as its restriction to parabolic subgroups. We prove that it is faithful when W is a Coxeter group of type ADE and odd dihedral types, and conjecture its faithfulness when W has a single class of reflections. If true, this conjecture would imply various group-theoretic properties for these groups, that we prove separately to be true for the other groups.

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arXiv2008

Knizhnik-Zamolodchikov bundles are topologically trivial

We prove that the vector bundles at the core of the Knizhnik-Zamolodchikov and quantum constructions of braid groups representations are topologically trivial bundles. We provide partial generalizations of this result to generalized braid groups. A crucial intermediate result is that the representation ring of the symmetric group on n letters is generated by the alternating powers of its natural n-dimensional representation.

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arXiv2008

Reflection groups acting on their hyperplanes

After having established elementary results on the relationship between a finite complex (pseudo-)reflection group W < GL(V) and its reflection arrangement A, we prove that the action of W on A is canonically related with other natural representations of W, through a `periodic' family of representations of its braid group. We also prove that, when W is irreducible, then the squares of defining linear forms for A span the quadratic forms on V, which imply |A| >= n(n+1)/2 for n = dim V, and relate the W-equivariance of the corresponding map with the period of our family.

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arXiv2008

Branching properties for the groups G(de,e,r)

We study general properties of the restriction of the representations of the finite complex reflection groups $G(de,e,r+1)$ to their maximal parabolic subgroups of type $G(de,e,r)$, and focus notably on the multiplicity of components. In combinatorial terms, this amounts to the following question : which symmetries arise or disappear when one changes (exactly) one pearl in a combinatorial necklace ?

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arXiv2008

Group algebras of finite groups as Lie algebras

We consider the natural Lie algebra structure on the (associative) group algebra of a finite group $G$, and show that the Lie subalgebras associated to natural involutive antiautomorphisms of this group algebra are reductive ones. We give a decomposition in simple factors of these Lie algebras, in terms of the ordinary representations of $G$.

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arXiv2008

Characters of the Grothendieck-Teichmueller group through rigidity of the Burau representation

We present examples of characters of absolute Galois groups of number fields that can be recovered through their action by automorphisms on the profinite completion of the braid groups, using a ``rigidity'' approach. The way we use to recover them is through classical representations of the braid groups, and in particular through the Burau representation. This enables one to extend these characters to Grothendieck-Teichmueller groups.

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arXiv2007

Cabling Burau Representation

The Burau representation enables to define many other representations of the braid group $B_n$ by the topological operation of ``cabling braids''. We show here that these representations split into copies of the Burau representation itself and of a representation of $B_n/(P_n,P_n)$. In particular, we show that there is no gain in terms of faithfulness by cabling the Burau representation.

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arXiv2006

On the representation theory of braid groups

This work presents an approach towards the representation theory of the braid groups $B_n$. We focus on finite-dimensional representations over the field of Laurent series which can be obtained from representations of infinitesimal braids, with the help of Drinfeld associators. We set a dictionary between representation-theoretic properties of these two structures, and tools to describe the representations thus obtained. We give an explanation for the frequent apparition of unitary structures on classical representations. We introduce new objects -- varieties of braided extensions, infinitesimal quotients -- which are useful in this setting, and analyse several of their properties. Finally, we review the most classical representations of the braid groups, show how they can be obtained by our methods and how this setting enrich our understanding of them.

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arXiv2006

Sur les representations de Krammer generiques

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.

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arXiv2006

L'algebre de Lie des transpositions

For any $n \geq 3$ we obtain the decomposition in simple factors of the Lie subalgebra of the group algebra of the symmetric group on $n$ letters generated by the transpositions. This enables us to determine the algebraic hull of the braid group $\B_n$ and of several of its subgroups inside the representations of the Iwahori-Hecke algebra of type A.

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arXiv2005

Monodromie algebrique des groupes d'Artin diedraux

Towards the study of the representation theory of any dihedral Artin group B, we build rational morphisms from B to the group of invertible elements of the associated infinitesimal braids algebra. For this we build analogues of Drinfeld associators, in rank 2 for arbitrary dihedral symmetries of the plane.

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arXiv2005

Caracteres de rigidite du groupe de Grothendieck-Teichmuller

Let $\k$ be a (topological) field of characteristic 0. Using a Drinfeld associator $Φ$, a representation $Φ(ρ)$ of the braid group over the field $\k((h))$ of Laurent series can be associated to any representation $ρ$ of a certain Hopf algebra $\mathfrak{B}_n(\k)$. We investigate the dependance in $Φ$ of $Φ(ρ)$ for a certain class of representations -- so-called GT-rigid representations -- and deduce from it (continuous) projective representations of the Grothendieck-Teichmuller group $GT_1(\k)$, hence for $\k = \Q_l$ representations of the absolute Galois group of $\Q(μ_{l^{\infty}})$. In most situations, these projective representations can be decomposed into linear characters, which we do for the representations of the Iwahori-Hecke algebra of type A. In this case, we moreover express $Φ(ρ)$ when $Φ$ is even, and get unitary matrix models for the representations of the Iwahori-Hecke algebra. With respect to the action of $GT_1(\k)$, the representations of this algebra corresponding to hook diagrams have noticeable properties.

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