Counting factorizations of Coxeter elements into products of reflections

Abstract

In this paper, we count factorizations of Coxeter elements in well-generated complex reflection groups into products of reflections. We obtain a simple product formula for the exponential generating function of such factorizations, which is expressed uniformly in terms of natural parameters of the group. In the case of factorizations of minimal length, we recover a formula due to P. Deligne, J. Tits and D. Zagier in the real case and to D. Bessis in the complex case. For the symmetric group, our formula specializes to a formula of D. M. Jackson.

38 pages, including 18 pages appendix. To appear in Journal of the London Mathematical Society. v3: minor changes and corrected references; v2: added extended discussion on the definition of Coxeter elements