Combinatorial Theory

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 \(G(m, p, n)\).

Mathematics Subject Classifications: 05A05, 05A15, 05E18, 20F55