Hurwitz numbers for reflection groups I: Generatingfunctionology

Abstract

The classical Hurwitz numbers count the fixed-length transitive transposition factorizations of a permutation, with a remarkable product formula for the case of minimum length (genus $0$). We study the analogue of these numbers for reflection groups with the following generalization of transitivity: say that a reflection factorization of an element in a reflection group $W$ is full if the factors generate the whole group $W$. We compute the generating function for full factorizations of arbitrary length for an arbitrary element in a group in the combinatorial family $G(m,p,n)$ of complex reflection groups in terms of the generating functions of the symmetric group ${\mathfrak{S}}_n$ and the cyclic group of order $m/p$. As a corollary, we obtain leading-term formulas which count minimum-length full reflection factorizations of an arbitrary element in $G(m,p,n)$ in terms of the Hurwitz numbers of genus $0$ and $1$ and number-theoretic functions. We also study the structural properties of such generating functions for any complex reflection group; in particular, we show via representation-theoretic methods that they can by expressed as finite sums of exponentials of the variable.

Submission includes file "data_file.txt"; 23 pages plus an Appendix; comments very much welcome!

References in corpus (2)

• The Hurwitz action in complex reflection groups
• Hurwitz numbers for reflection groups II: Parabolic quasi-Coxeter elements

Cited by in corpus (3)

• Hurwitz numbers for reflection groups II: Parabolic quasi-Coxeter elements
• Hurwitz numbers for reflection groups III: Uniform formulas
• In which it is proven that, for each parabolic quasi-Coxeter element in a finite real reflection group, the orbits of the Hurwitz action on its reflection factorizations are distinguished by the two obvious invariants