Hurwitz numbers for reflection groups II: Parabolic quasi-Coxeter elements

Abstract

We define parabolic quasi-Coxeter elements in well generated complex reflection groups. We characterize them in multiple natural ways, and we study two combinatorial objects associated with them: the collections ${\mathrm{Red}}_W(g)$ of reduced reflection factorizations of $g$ and $\mathrm{RGS}(W,g)$ of the relative generating sets of $g$. We compute the cardinalities of these sets for large families of parabolic quasi-Coxeter elements and, in particular, we relate the size $\#{\mathrm{Red}}_W(g)$ with geometric invariants of Frobenius manifolds. This paper is second in a series of three; we will rely on many of its results in part III to prove uniform formulas that enumerate full reflection factorizations of parabolic quasi-Coxeter elements, generalizing the genus-$0$ Hurwitz numbers.

v2: 50 pages, minor edits, comments very much welcome!

References in corpus (3)

• Enumerative properties of generalized associahedra
• On classification and construction of algebraic Frobenius manifolds
• On cycle decompositions in Coxeter groups

Cited by in corpus (3)

• Coincidences between intervals in two partial orders on complex reflection groups
• Hurwitz numbers for reflection groups III: Uniform formulas
• In which it is proven that, for each parabolic quasi-Coxeter element in a finite real reflection group, the orbits of the Hurwitz action on its reflection factorizations are distinguished by the two obvious invariants