Sur les représentations de Krammer génériques

Abstract

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.

Braids inside the Birman–Wenzl–Murakami algebra

“…Here we exhibit a reductive Lie subalgebra of the Brauer centralizer algebra that plays a similar role, and that we decompose accordingly. A consequence is the following, which generalizes [14,Theorem 2]. Recall from [3] that BMW n .s;˛/ is split semisimple over K (see Birman and Wenzl [3,Theorem 3.7], the proof given there being valid over Q.s;˛/, not only C.s;˛/).…”

“…s ij , p ij 7 ! p ij of Br n .m/), which has been studied separately in [14], or OE∅ 4 (see Figure 3). In this last case, its restriction to Br 3 .m/ is the irreducible representation OE1 3 which is a Krammer representation, hence …”

“…For˛; s generic (say, algebraically independent over Q K ), this problem was solved by the author in [13] for the Hecke algebra representations and in [14] for the Krammer representation. The present paper is thus a sequel of these two previous works.…”

“…For the Hecke algebra representations [12], one can explicitly compute the matrix models obtained this way, which turn out to be the same than the ones obtained earlier by Wenzl, and check that they are indeed convergent. Another way, that we already used in [14] for the Krammer representation, is to approximate the representation over ROEOEh by equivalent representations over the ring of convergent power series. This is the device we use here, giving in the above spirit a new proof of the following result, which was originally proved by Wenzl [18] by using the Jones construction.…”

Braid groups and Artin groups

“…It is shown in [120] that: if Γ is of type A n , D n , E k (k = 6, 7, 8), then the image ofΦ is Zariski dense in GL(V). In particular, this shows thatΦ is irreducible (see also [154], [121], [50]). …”