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

Theo Douvropoulos, Joel Brewster Lewis, Alejandro H. Morales

TL;DR

This paper develops the theory of parabolic quasi-Coxeter elements in well-generated complex reflection groups, introducing reduced reflection factorizations Red_W(g) and relative generating sets RGS(W,g) and linking them to Hurwitz actions and Frobenius-manifold geometry. It provides structural characterizations, a unique-cycle decomposition, and transitivity insights (notably in the real case) while delivering extensive enumerative results for reduced factorizations and RGS across key infinite families, including explicit formulas in G(m,1,n) and G(m,m,n). The work also connects these combinatorial counts to geometric invariants via Frobenius manifolds and the Lyashko–Looijenga map, setting the stage for Part III’s uniform full-factorization formulas that generalize genus-0 Hurwitz numbers. Finally, it lays out a detailed framework for counting relative generating sets and discusses broader implications, conjectures, and open questions in real and complex reflection-group theory.

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 $\operatorname{Red}_W(g)$ of reduced reflection factorizations of $g$ and $\operatorname{RGS}(W,g)$ 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 $\#\operatorname{Red}_W(g)$ 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.

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

TL;DR

This paper develops the theory of parabolic quasi-Coxeter elements in well-generated complex reflection groups, introducing reduced reflection factorizations Red_W(g) and relative generating sets RGS(W,g) and linking them to Hurwitz actions and Frobenius-manifold geometry. It provides structural characterizations, a unique-cycle decomposition, and transitivity insights (notably in the real case) while delivering extensive enumerative results for reduced factorizations and RGS across key infinite families, including explicit formulas in G(m,1,n) and G(m,m,n). The work also connects these combinatorial counts to geometric invariants via Frobenius manifolds and the Lyashko–Looijenga map, setting the stage for Part III’s uniform full-factorization formulas that generalize genus-0 Hurwitz numbers. Finally, it lays out a detailed framework for counting relative generating sets and discusses broader implications, conjectures, and open questions in real and complex reflection-group theory.

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 of reduced reflection factorizations of and of the relative generating sets of . We compute the cardinalities of these sets for large families of parabolic quasi-Coxeter elements and, in particular, we relate the size 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- Hurwitz numbers.
Paper Structure (44 sections, 51 theorems, 83 equations, 1 figure, 7 tables)

This paper contains 44 sections, 51 theorems, 83 equations, 1 figure, 7 tables.

Table of Contents

  1. Introduction
  2. Structural results
  3. Enumerative results
  4. Towards counting full factorizations of parabolic quasi-Coxeter elements
  5. Summary
  6. Reflection groups, good generating sets, and parabolic subgroups
  7. Real reflection groups
  8. Complex reflection groups
  9. Well generated groups and good generating sets
  10. Reflection factorizations and reflection length
  11. Parabolic subgroups
  12. The Hurwitz action
  13. Parabolic quasi-Coxeter elements
  14. Origins of (parabolic) quasi-Coxeter elements
  15. Conjugacy classes of Weyl groups
  16. ...and 29 more sections

Key Result

Theorem 1.1

Let $W \leq {\mathrm{GL}}(V)$ be a Weyl group of rank $n$, $g$ an element of $W$, and $W_g$ the parabolic closure of $g$. Then the following statements are equivalent.

Figures (1)

  • Figure 1: The relative tree representing the relative generating set $\{(1,11),(2,6),(2,4),(2,12)\}$ of $\sigma=(1\, 5\, 8)(2)(3\, 12\,9)(4\,7\,10)(6\,11)$ and its underlying vertex-labeled tree (left) and the edge-labeled cactus appearing in the work Duchi--Poulhalhon--Schaeffer and its underlying edge-labeled tree (right). The underlying labeled trees are in correspondence by selecting a vertex in the edge-labeled tree (illustrated with a larger vertex) to be labeled $1$ and pushing away the label $i$ of an edges to a vertex label $i+1$.

Theorems & Definitions (115)

  • Theorem 1.1
  • Theorem 1.1: characterization of parabolic quasi-Coxeter elements
  • Proposition 1.0
  • Theorem 1.1
  • Theorem
  • Definition 2.1: good generating sets
  • Proposition 2.2: BW
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • ...and 105 more