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Characteristic Polynomials of Deformations of Coxeter Arrangements via levels of regions

Ningxin Zhang

Abstract

We obtain a novel formula for characteristic polynomials of deformations of the Braid arrangement using the notion of levels of regions. As an application, we recover and strengthen results of Chen et al. on the characteristic polynomial of several specific types of hyperplane arrangements via much simpler arguments. Our theorem also generalizes to type B.

Characteristic Polynomials of Deformations of Coxeter Arrangements via levels of regions

Abstract

We obtain a novel formula for characteristic polynomials of deformations of the Braid arrangement using the notion of levels of regions. As an application, we recover and strengthen results of Chen et al. on the characteristic polynomial of several specific types of hyperplane arrangements via much simpler arguments. Our theorem also generalizes to type B.

Paper Structure

This paper contains 8 sections, 9 theorems, 47 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Characteristic polynomials
  4. Levels of regions
  5. Expanding $\chi(t)$ to the numbers of regions with different levels
  6. Deformations of type $A$ Coxeter arrangements
  7. Deformations of type $B$ Coxeter arrangements
  8. Applications of main results

Key Result

Theorem 1.1

Let $\mathcal{A}$ be a non-degenerate deformation of $\operatorname{Cox}_{A}(n)$ as in typeA. Then where $r_k(\mathcal{A})$ is the number of regions with level $k$ in arrangement $\mathcal{A}$.

Figures (3)

  • Figure 1: Examples of deformations of Coxeter arrangements.
  • Figure 2: The hyperplane arrangement $\mathcal{A}$
  • Figure 3: The regions increased after adding a hyperplane.

Theorems & Definitions (20)

  • Theorem 1.1
  • Remark 1.2
  • Example 1.3
  • Theorem 1.4
  • Remark 1.5
  • Example 1.6
  • Definition 2.1
  • Lemma 2.2: Deletion-restriction stanley2004introduction
  • Theorem 2.3: athanasiadis96
  • Theorem 2.4: Zaslavsky Theorem
  • ...and 10 more