Ivan Marin

THE CENTER OF PURE COMPLEX BRAID GROUPS

Abstract Broue, Malle and Rouquier (1998) conjectured in [2] that the center of the pure braid group of an irreducible finite complex reflection group is cyclic. We prove this conjecture, for the remaining exceptional types, using the analogous result for the full braid group due to Bessis, and we actually prove the stronger statement that any finite index subgroup of such braid group has cyclic center.

Caractères de rigidité du groupe de Grothendieck–Teichmüller

— Let k be a (topological) field of characteristic 0. Using a Drinfeld associator, a representation b Φ(ρ) of the braid group over the field k((h)) of Laurent series can be associated to any representation of a certain Hopf algebra Bn(k). We investigate the dependance in Φ of b 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 GT1(k), hence for k = Ql representations of the absolute Galois group of Q(μl∞). 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 b 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 GT1(k), the representations of this algebra corresponding to hook diagrams have noticeable properties.

Caractérisations de la représentation de Burau

Abstract This paper is devoted to characterizations of the (reduced) Burau representation of the Artin braid group, in terms of rigid local systems. We prove that the Burau representation is the only representation of the Hecke algebra for which some local system associated to every linear representation of the braid group is irreducible and rigid in the sense of Katz. We also use previous results to give a characterization of the corresponding Knizhnik-Zamolodchikov system.

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.

Éléments de Jucys–Murphy Généralisés

ABSTRACT We define Jucys–Murphy elements for the finite Coxeter groups which do not contain D 4 as a parabolic subgroup. We prove that these elements share some previously established properties with the original Jucys–Murphy elements of the symmetric group. This enables one to envisage an approach to the representation theory of these groups similar to the Vershik-Okounkov reconstruction for the symmetric group. Nous généralisons les éléments de Jucys–Murphy aux groupes de Coxeter finis qui ne contiennent pas D 4 comme sous-groupe parabolique. Nous montrons que ces éléments vérifient certaines propriétés établies précédemment pour le groupe symétrique, et permettent ainsi d’envisager une approche des représentations de ces groupes à la manière de Vershik et Okounkov pour le groupe symétrique.