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Hurwitz numbers for reflection groups III: Uniform formulas

Theo Douvropoulos, Joel Brewster Lewis, Alejandro H. Morales

TL;DR

This work develops uniform product formulas for counting minimum-length full reflection factorizations $F_W^{\mathrm{full}}(g)$ of parabolic quasi-Coxeter elements $g$ in Weyl groups and well-generated complex reflection groups, extending genus-0 Hurwitz numbers. The Weyl-case formula factors as a product over generalized cycles and includes the ratio of connection indices $I(W_g)/I(W)$ and the relative generating-set count $\#\mathrm{RGS}(W,g)$, while the complex-case formula sums over $\mathrm{RGS}(W,g)$ weighted by the Grammian determinant $\mathrm{GD}$. In the symmetric group limit $W=\mathfrak{S}_n$, the results recover the classical genus-0 Hurwitz numbers $H_0(\lambda)$, linking the reflection-group framework to classical enumerative topology. The paper combines case-by-case analyses for $G(m,1,n)$ and $G(m,m,n)$ with a Weyl-cut-and-join recursion and computer-assisted verifications for exceptional groups, and discusses connections to geometry, weighted Hurwitz theory, and potential uniform proofs beyond case-splitting.

Abstract

We give uniform formulas for the number of full reflection factorizations of a parabolic quasi-Coxeter element in a Weyl group or complex reflection group, generalizing the formula for the genus-0 Hurwitz numbers. This paper is the culmination of a series of three.

Hurwitz numbers for reflection groups III: Uniform formulas

TL;DR

This work develops uniform product formulas for counting minimum-length full reflection factorizations of parabolic quasi-Coxeter elements in Weyl groups and well-generated complex reflection groups, extending genus-0 Hurwitz numbers. The Weyl-case formula factors as a product over generalized cycles and includes the ratio of connection indices and the relative generating-set count , while the complex-case formula sums over weighted by the Grammian determinant . In the symmetric group limit , the results recover the classical genus-0 Hurwitz numbers , linking the reflection-group framework to classical enumerative topology. The paper combines case-by-case analyses for and with a Weyl-cut-and-join recursion and computer-assisted verifications for exceptional groups, and discusses connections to geometry, weighted Hurwitz theory, and potential uniform proofs beyond case-splitting.

Abstract

We give uniform formulas for the number of full reflection factorizations of a parabolic quasi-Coxeter element in a Weyl group or complex reflection group, generalizing the formula for the genus-0 Hurwitz numbers. This paper is the culmination of a series of three.
Paper Structure (30 sections, 29 theorems, 104 equations, 2 figures)

This paper contains 30 sections, 29 theorems, 104 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main theorems
  3. Recovering the original Hurwitz formula for the symmetric group
  4. Overview of the paper
  5. Preliminaries and background
  6. Real reflection groups
  7. Complex reflection groups
  8. Root systems
  9. Reflection length and full reflection length
  10. Coxeter elements and quasi-Coxeter elements
  11. Parabolic subgroups and parabolic quasi-Coxeter elements
  12. Reflection generating sets
  13. Proof of the main results
  14. The main theorems are equivalent for Weyl groups
  15. Reduction to the irreducible case
  16. ...and 15 more sections

Key Result

Theorem 1.1

The minimum length of a transitive transposition factorization in $\mathfrak{S}_n$ of a permutation of cycle type $\lambda:=(\lambda_1,\ldots,\lambda_r)$ is $n+r-2$. The number of such factorizations is In particular, $H_0(1^n) = (2n-2)!\cdot n^{n-3}$ and $H_0(n) = n^{n-2}$.

Figures (2)

  • Figure 1: At left, the hyperplane arrangement of $B_2$ labeled by reflections (as signed permutations in one-line notation; in black) and the subgroups they generate (in blue). At right, the poset of pairs $(g_i,W_i)$ as in Section \ref{['sec:poset']}, whose maximal chains correspond to minimum-length full reflection factorizations of the identity in $B_2$. The longest element $-1 = w_0 = \bar{1}\bar{2}$ (which is not parabolic quasi-Coxeter in $B_2$) and the two Coxeter elements $\bar{2}1$ and $2\bar{1}$ appear in the middle rank.
  • Figure 2: Illustration of the quantities $\xi+\overline{\xi}$ over roots of unity appearing in the identity of Proposition \ref{['prop: primitive root identity 2']} for $m=2, 6, 9$. The primitive roots are denoted in black ($\bullet$) while the other roots are denoted in red ($\bullet$). The root $\xi$ is paired with $\overline{\xi}$ by dashed lines.

Theorems & Definitions (49)

  • Theorem 1.1: Hurwitz formula Hurwitz
  • Theorem 1.1: main theorem for Weyl groups
  • Corollary 1.1
  • Theorem 1.1: main theorem for complex reflection groups
  • Corollary 2.1: DLM2
  • Proposition 2.2: part of DLM1
  • Corollary 2.3: DLM2
  • Proposition 2.4: LW
  • Theorem 2.5: Steinberg's theorem Steinberg
  • Proposition 2.6: taylor
  • ...and 39 more