Explicit Presentations for Exceptional Braid Groups

To any complex reflection group W ⊂ GL(V ), one may attach a braid group B(W ), defined as the fundamental group of the space of regular orbits for the action of W on V [Broué et al. 98]. The “ordinary” braid group on n strings, introduced by [Artin 47], corresponds to the case of the symmetric group Sn, in its monomial reflection representation in GLn(C). More generally, any Coxeter group can be seen as a complex reflection group, by complexifying the reflection representation. It is proved in [Brieskorn 71] that the corresponding braid group can be described by an Artin presentation, obtained by “forgetting” the quadratic relations in the Coxeter presentation. Many geometric properties of Coxeter groups still hold for arbitrary complex reflection groups. Various authors, including Coxeter himself, have described “Coxeter-like” presentations for complex reflection groups. Of course, one would like to have not just a “Coxeter-like” presentation for W , but also an “Artin-like” presentation for B(W ). The problem can be reduced to the irreducible case. Irreducible complex reflection groups have been classified by [Shephard et al. 54]: there is an infinite family G(de, e, r) (which contains the infinite families of Coxeter groups), plus 34 exceptional groups G4, . . . , G37 (among them are the exceptional Coxeter groups). Before this note, presentations were known for all but 6 exceptional groups (see the tables of [Broué et al. 98]):