Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Computing characteristic polynomials of hyperplane arrangements with symmetries

Taylor Brysiewicz, Holger Eble, Lukas Kühne

TL;DR

The paper addresses the computational challenge of enumerating chambers and computing the characteristic polynomial $\chi_{\mathcal{A}}(t)$ of hyperplane arrangements. It introduces a symmetry-aware deletion–restriction algorithm and implements it in CountingChambers.jl on top of OSCAR to exploit automorphisms of the arrangement, yielding exact, scalable calculations even for highly symmetric cases. The authors demonstrate substantial speedups and provide extensive results for classical families (e.g., threshold and resonance arrangements) and discriminantal configurations, informing both combinatorial geometry and applications in physics. The work delivers a practical tool for reliable chamber counts and refined Whitney numbers, with potential impact on related areas requiring exact arithmetic and symmetry exploitation.

Abstract

We introduce a new algorithm computing the characteristic polynomials of hyperplane arrangements which exploits their underlying symmetry groups. Our algorithm counts the chambers of an arrangement as a byproduct of computing its characteristic polynomial. We showcase our julia implementation, based on OSCAR, on examples coming from hyperplane arrangements with applications to physics and computer science.

Computing characteristic polynomials of hyperplane arrangements with symmetries

TL;DR

The paper addresses the computational challenge of enumerating chambers and computing the characteristic polynomial of hyperplane arrangements. It introduces a symmetry-aware deletion–restriction algorithm and implements it in CountingChambers.jl on top of OSCAR to exploit automorphisms of the arrangement, yielding exact, scalable calculations even for highly symmetric cases. The authors demonstrate substantial speedups and provide extensive results for classical families (e.g., threshold and resonance arrangements) and discriminantal configurations, informing both combinatorial geometry and applications in physics. The work delivers a practical tool for reliable chamber counts and refined Whitney numbers, with potential impact on related areas requiring exact arithmetic and symmetry exploitation.

Abstract

We introduce a new algorithm computing the characteristic polynomials of hyperplane arrangements which exploits their underlying symmetry groups. Our algorithm counts the chambers of an arrangement as a byproduct of computing its characteristic polynomial. We showcase our julia implementation, based on OSCAR, on examples coming from hyperplane arrangements with applications to physics and computer science.

Paper Structure

This paper contains 23 sections, 4 theorems, 15 equations, 10 figures, 8 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Hyperplane arrangements
  3. The characteristic polynomial
  4. A deletion-restriction algorithm
  5. A simple deletion-restriction algorithm
  6. Computationally representing deletions and restrictions
  7. Automorphisms of hyperplane arrangements
  8. Enumeration algorithm with symmetry
  9. Representing orbits
  10. Accumulating the Whitney numbers and skipping levels
  11. Relation to OSCAR
  12. Functionality of CountingChambers.jl
  13. Examples and integer sequences
  14. Separability arrangements
  15. The threshold arrangement
  16. ...and 8 more sections

Key Result

Lemma 5

Given a hyperplane $H \in \mathcal{A}$, we have that $\chi_{\mathcal{A}}(t) = \chi_{\mathcal{A} \backslash \{H\} }(t) - \chi_{\mathcal{A}^H}(t).$ In particular, $b(\mathcal{A}) = b(\mathcal{A}\backslash \{H\}) + 0|b(\mathcal{A}^{H})$ where $0|b$ means prepending the vector $b$ with a zero.

Figures (10)

  • Figure 1: The arrangement introduced in \ref{['ex:arr']}.
  • Figure 2: The intersections of a generic flag (purple) in ${\mathbb{R}}^2$ with the chambers of $\mathcal{A}$. The point $F_0$ intersects one chamber, $F_1$ intersects four others, and $F_2$ intersects the remaining $5$, and so $b(\mathcal{A}) = (1,4,5)$.
  • Figure 3: The tree structure of Algorithm \ref{['algo:simpleDR']} on the hyperplane arrangement from Example \ref{['ex:arr']}. Hyperplanes are chosen (Line \ref{['chooseHyperplane_Alg1']}) according to the ordering $\{1,2,3,4\}$. In each box, the ambient space of the arrangement is shaded green. Deletions are marked with red edges (left children) and restrictions with blue edges (right children). Each arrangement box has the Whitney numbers above its upper right corner.
  • Figure 4: Whitney numbers via simple deletion and restriction
  • Figure 5: Whitney numbers via extended deletion and restriction
  • ...and 5 more figures

Theorems & Definitions (17)

  • Definition 1
  • Example 2
  • Definition 3
  • Example 4
  • Lemma 5: OT92
  • Lemma 6
  • proof
  • Definition 7
  • Remark 8
  • Lemma 9
  • ...and 7 more