# Nikolaus Conference 2018

Speaker: Georges Neaime (Bielefeld)

Title: Towards the Linearity of the Complex Braid Groups

Abstract:

Both Bigelow and Krammer solved a long-standing open problem about the linearity of the braid groups. Shortly thereafter, Krammer's representation as well as Krammer's faithfulness proof have been extended to all finite-type Artin-Tits groups by a work of Cohen-Wales and Digne. Moreover, many subsequent discoveries unveiled a link between these representations and the BMW (Birman-Murakami-Wenzl) algebras. One would like to extend these properties to the complex braid groups constructed by Broué-Malle-Rouquier for each complex reflection group. These objects are a generalization of the finite-type Artin-Tits groups. Note that the complex braid groups attached to the general series of complex reflection groups belong to two 2-parameter families denoted by $$B(e,e,n)$$ and $$B(2e,e,n)$$. It is already known that the groups $$B(2e,e,n)$$ are linear. This therefore arises the natural question whether the groups $$B(e,e,n)$$ are also linear. A positive answer for this question is conjectured to be true by Marin who (analytically) constructed a generalization of the Krammer's representations for the case of the complex braid groups. In this talk, we provide a basic background material and establish several properties that synthesize our definition of a BMW and Brauer algebras associated with $$B(e,e,n)$$. We also construct Krammer representations for some cases of $$B(e,e,n)$$. Throughout the talk, we propose several conjectures about these representations and about the BMW and Brauer algebras associated with $$B(e,e,n)$$. We hope that this will help for further progress in the subject of the linearity of the complex braid groups.