Nikolaus Conference 2018

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.