Factorizations of Coxeter Elements in Complex Reflection Groups

In [CS12], Chapuy and Stump studied factorizations of Coxeter elements into products of reflections in well-generated, irreducible complex reflection groups, giving a simple closed form expression for their exponential generating function depending on certain natural parameters. In this work, their methods are used to consider a more general multivariate generating function FACW (u1, . . . , u`) in which the number of reflections from each hyperplane orbit is recorded. This generating function takes a similar simple form which can be stated uniformly for well-generated irreducible complex reflection groups in terms of certain data associated to orbits of reflecting hyperplanes, and specializes to that of Chapuy and Stump. A consequence of this more refined generating function – that the Coxeter number of any well-generated complex reflection group W is determined by certain data associated to any W -orbit of reflecting hyperplanes– is stated, and a case-free argument is given for real reflection groups.