Hurwitz numbers for reflection groups I: generatingfunctionology
Theo Douvropoulos, Joel Brewster Lewis, Alejandro H. Morales · Enumerative Combinatorics and Applications · 2022
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 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 be expressed as finite sums of exponentials of the variable.