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 Sn 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.