Presentations for crystallographic complex reflection groups

Abstract

We find presentations for the irreducible crystallographic complex reflection groupsW whose linear part is not the complexification of a real reflection group. The presentations are given in the form of graphs resembling Dynkin diagrams and very similar to the presentations for finite complex reflection groups given in [2]. As in the case of affine Weyl groups, they can be obtained by adding a further node to the diagram for the linear part. We then classify the reflections in the groupsW and the minimal number of them needed to generateW, using the diagrams. Finally we show for more than half of the infinite series that a presentation for the fundamental group of the space of regular orbits ofW can be derived from our presentations.