# Combinatorics and Algebra-Hurwitz numbers for reflection groups

Title: Hurwitz numbers for reflection groups

Speaker: Joel Lewis, GWU

Date and time: Friday, December 2, 4–5 pm

Place: Rome 204

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. For the combinatorial families, we compute the generating function for full factorizations of arbitrary length for arbitrary elements, in terms of the generating functions of the symmetric group and an appropriate cyclic group. As a corollary, we obtain leading-term formulas that count minimum-length full reflection factorizations of arbitrary elements in terms of the Hurwitz numbers of genus 0 and 1 and number-theoretic functions. In the case of an arbitrary complex reflection group, we give uniform (i.e., case-free) formulas for the number of genus-0 full reflection factorizations of a large family of elements (the parabolic quasi-Coxeter elements).