Hurwitz Orbits of Equal Size

TL;DR

This paper investigates conditions under which different factorizations in groups have Hurwitz orbits of equal size, providing theoretical results and applications to complex reflection groups.

Contribution

It introduces new criteria for equal Hurwitz orbit sizes, including operations like cycling, flipping, inverting, and double reverse factorizations, with applications to complex reflection groups.

Findings

Cycling elements preserves Hurwitz orbit size.

Flipping and inverting elements preserve Hurwitz orbit size.

Double reverse factorizations have equal Hurwitz orbit sizes in certain groups.

Abstract

We provide a variety of cases in which two factorizations have Hurwitz orbits of the same size. We begin with prototypical results about factorizations of length two, and show that cycling elements or flipping and inverting elements in any factorization preserves Hurwitz orbit size. We prove that "double reverse" factorizations in groups with special presentations have Hurwitz orbits of equal size, and end with applications to complex reflection groups.