The Hurwitz action in complex reflection groups

TL;DR

This paper studies the Hurwitz action on reflection factorizations in complex reflection groups, providing enumeration, transitivity criteria, and characterizations of special elements like quasi-Coxeter elements.

Contribution

It introduces a comprehensive enumeration of Hurwitz orbits, characterizes transitivity conditions, and identifies quasi-Coxeter elements in the family G(m, p, n).

Findings

Enumerated Hurwitz orbits for shortest reflection factorizations.

Provided criteria for transitivity of the Hurwitz action.

Characterized quasi-Coxeter elements in G(m, p, n).

Abstract

We enumerate Hurwitz orbits of shortest reflection factorizations of an arbitrary element in the infinite family $G(m, p, n)$ of complex reflection groups. As a consequence, we characterize the elements for which the action is transitive and give a simple criterion to tell when two shortest reflection factorizations belong to the same Hurwitz orbit. We also characterize the quasi-Coxeter elements (those with a shortest reflection factorization that generates the whole group) in $G(m, p, n)$.