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A numerical characterization of Dunkl systems

Martin de Borbon, Dmitri Panov

TL;DR

The paper provides a numerical characterization of Dunkl metrics for weighted hyperplane arrangements by establishing a variational principle: Dunkl metrics arise as extremizers of the Hirzebruch quadratic form $Q$ on the weight vector $\mathbf{a}$. It proves that stability of $(\mathcal{H},\mathbf{a})$ together with $Q(\mathbf{a})=0$ is equivalent to the existence of a Dunkl metric, and further shows this is equivalent to a system of linear equalities $s_i(\mathbf{a}) = \frac{d-1}{d}\sum_j a_j$, turning the problem into a polyhedral intersection: the interior of the cone over the matroid polytope of $\mathcal{H}$ with a linear subspace in $\mathbb{R}^n$. The approach relies on a tight-frame inequality to derive the quadratic bound and translates equality into the Dunkl commutation relations, providing a concise, combinatorial criterion for Dunkl realizability. The results connect stability, matroid geometry, and harmonic analysis on hyperplane arrangements, with implications for understanding Dunkl connections in low and high dimensions.

Abstract

We give a numerical characterization of weighted hyperplane arrangements arising from Dunkl systems.

A numerical characterization of Dunkl systems

TL;DR

The paper provides a numerical characterization of Dunkl metrics for weighted hyperplane arrangements by establishing a variational principle: Dunkl metrics arise as extremizers of the Hirzebruch quadratic form on the weight vector . It proves that stability of together with is equivalent to the existence of a Dunkl metric, and further shows this is equivalent to a system of linear equalities , turning the problem into a polyhedral intersection: the interior of the cone over the matroid polytope of with a linear subspace in . The approach relies on a tight-frame inequality to derive the quadratic bound and translates equality into the Dunkl commutation relations, providing a concise, combinatorial criterion for Dunkl realizability. The results connect stability, matroid geometry, and harmonic analysis on hyperplane arrangements, with implications for understanding Dunkl connections in low and high dimensions.

Abstract

We give a numerical characterization of weighted hyperplane arrangements arising from Dunkl systems.
Paper Structure (7 sections, 4 theorems, 18 equations)

This paper contains 7 sections, 4 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Stability and the Hirzebruch quadratic form
  4. Tight frames
  5. Main results
  6. Quadratic inequality
  7. Linear constraints

Key Result

Proposition 2.2

Suppose that $(\mathcal{H}, \mathbf{a})$ is stable. Then there exists a -unique up to scale- Hermitian inner product $\langle \cdot , \cdot \rangle$ on $\mathbb{C}^d$ such that where $P_i$ is the orthogonal projection to $H_i^{\perp}$ with respect to $\langle \cdot , \cdot \rangle$ and $c \in \mathbb{R}$.

Theorems & Definitions (15)

  • Definition 1.1
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3: borbonpanov
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6
  • Remark 2.7
  • Theorem 3.1
  • ...and 5 more