On Coxeter Diagrams of complex reflection groups

We study complex Coxeter diagrams of some unitary reflection groups. Using solely the combinatorics of diagrams, we give a new proof of the classification of root lattices defined over E = Z[e2πi/3]: there are only four such lattices, namely, the E–lattices whose real forms are A2, D4, E6 and E8. Next, we address the issue of characterizing the diagrams for unitary reflection groups, a question that was raised by Broue, Malle and Rouquier. To this end, we describe an algorithm which, given a unitary reflection group G, picks out a set of complex reflections. If G is the reflection group of a root lattice defined over E , the reflections selected by the algorithm form the known diagram for G. For the reflection group of the complex Coxeter-Todd lattice KE 12, we find a new diagram that extends to an “affine diagram” with Z/7Z symmetry. If G is a Weyl group, the algorithm immediately yields a set of simple roots. Otherwise, experimental evidences indicate that the algorithm selects a minimal generating set of reflections if G is primitive and G has a set of roots whose Z–span is a discrete subset of the ambient vector space.