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The geometry of a counting formula for deformations of the braid arrangement

Neha Goregaokar, Aaron Lin

Abstract

We consider real hyperplane arrangements whose hyperplanes are of the form $\{x_i - x_j = s\}$ for some integer $s$, which we call deformations of the braid arrangement. In 2018, Bernardi gave a counting formula for the number of regions of any deformation of the braid arrangement $\mathcal{A}$ as a signed sum over some decorated trees. He further showed that each of these decorated trees can be associated to a region $R$ of the arrangement $\mathcal{A}$, and hence we can consider the contribution of each region to the signed sum. Bernardi also implicitly showed that for transitive arrangements, the contribution of any region of the arrangement is $1$. We remove the transitivity condition, showing that for any deformation of the braid arrangement the contribution of a region to the signed sum is $1$. This provides an alternative proof of the original counting formula, and sheds light on the geometry underlying the formula. We further use this new geometric understanding to better understand the contribution of a tree.

The geometry of a counting formula for deformations of the braid arrangement

Abstract

We consider real hyperplane arrangements whose hyperplanes are of the form for some integer , which we call deformations of the braid arrangement. In 2018, Bernardi gave a counting formula for the number of regions of any deformation of the braid arrangement as a signed sum over some decorated trees. He further showed that each of these decorated trees can be associated to a region of the arrangement , and hence we can consider the contribution of each region to the signed sum. Bernardi also implicitly showed that for transitive arrangements, the contribution of any region of the arrangement is . We remove the transitivity condition, showing that for any deformation of the braid arrangement the contribution of a region to the signed sum is . This provides an alternative proof of the original counting formula, and sheds light on the geometry underlying the formula. We further use this new geometric understanding to better understand the contribution of a tree.

Paper Structure

This paper contains 10 sections, 14 theorems, 24 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Hyperplane Arrangements
  4. The Bernardi Formula
  5. Restriction to a region
  6. The Contribution of a Region
  7. Graphical Arrangements
  8. General Case
  9. The Contribution of a Tree
  10. Acknowledgments

Key Result

Theorem 2.8

Let $\mathbf{S} = (S_{a,b})_{1 \leq a < b \leq n}$ be a collection of finite sets of integers, and let $\mathcal{A}_{\mathbf{S}}$ be the $\mathbf{S}$-braid arrangement in $\mathbb{R}^n$. The number of regions of $\mathcal{A}_{\mathbf{S}}$ is given by where $|B|$ is the number of boxes in the $\mathbf{S}$-boxed tree $(T, B)$ and $\mathcal{U}_{\mathbf{S}}(n)$ denotes the set of $\mathbf{S}$-boxed t

Figures (9)

  • Figure 1: Braid arrangement for $n=3$. Each line corresponds to an equation of the form $x_i=x_j$.
  • Figure 2: A tree in $\mathcal{T}^{(2)}(7)$.
  • Figure 3: An example for an $\mathbf{S}$-boxed tree for $S_{a,b} = \{-2,1\}$ for all $a<b$.
  • Figure 4: Bijection between the set of trees in $\mathcal{T}^{(1)}(3)$ and the regions of the 1-Catalan arrangement corresponding to each tree.
  • Figure 5: Bijection between faces of $\mathcal{B}_3$ and ordered partitions of $[3]$.
  • ...and 4 more figures

Theorems & Definitions (57)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • Example 2.7
  • Theorem 2.8: Bernardi Formula for $\mathbf{S}$-braid arrangements BERNARDI2018466
  • Theorem 2.9: BERNARDI2018466
  • Definition 2.10
  • ...and 47 more