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Bijections for faces of braid-type arrangements

Olivier Bernardi

TL;DR

This work develops a general bijective framework that encodes faces of braid-type hyperplane arrangements in $\mathbb{R}^n$, defined by hyperplanes $\{x_i-x_j=s\}$, under a strong transitivity condition. The core construction is a bijection between faces and decorated plane trees (marked $(m,n)$-trees), with the face codimension equalling the number of marked edges; this extends and unifies Levear’s Catalan/Shi bijections and builds on a prior transitive-arrangement framework. The paper then derives generating functions for faces in symmetric transitive settings, including explicit equations for Catalan and semiorder cases and a transfer-matrix approach for arbitrary symmetric sets $S$. Additional sections illustrate the framework via multi-Catalan arrangements and interpolations between Catalan and Shi, and provide a detailed proof via a commutative diagram that connects restriction arrangements to transitive-tree encodings. Overall, the results give a robust, computationally tractable combinatorial description of faces in a broad family of braid-type arrangements with potential applications in enumeration and structural analysis.

Abstract

We establish a general bijective framework for encoding faces of some classical hyperplane arrangements. Precisely, we consider hyperplane arrangements in $\mathbb{R}^n$ whose hyperplanes are all of the form $\{x_i-x_j=s\}$ for some $i,j\in[n]$ and $s\in \mathbb{Z}$. Such an arrangement $A$ is \emph{strongly transitive} if it satisfies the following condition: if $\{x_i-x_j=s\}\notin A$ and $\{x_j-x_k=t\}\notin A$ for some $i,j,k\in [n]$ and $s,t\geq 0$, then $\{x_i-x_k=s+t\}\notin A$. For any strongly transitive arrangement $A$, we establish a bijection between the faces of $A$ and some set of decorated plane trees.

Bijections for faces of braid-type arrangements

TL;DR

This work develops a general bijective framework that encodes faces of braid-type hyperplane arrangements in , defined by hyperplanes , under a strong transitivity condition. The core construction is a bijection between faces and decorated plane trees (marked -trees), with the face codimension equalling the number of marked edges; this extends and unifies Levear’s Catalan/Shi bijections and builds on a prior transitive-arrangement framework. The paper then derives generating functions for faces in symmetric transitive settings, including explicit equations for Catalan and semiorder cases and a transfer-matrix approach for arbitrary symmetric sets . Additional sections illustrate the framework via multi-Catalan arrangements and interpolations between Catalan and Shi, and provide a detailed proof via a commutative diagram that connects restriction arrangements to transitive-tree encodings. Overall, the results give a robust, computationally tractable combinatorial description of faces in a broad family of braid-type arrangements with potential applications in enumeration and structural analysis.

Abstract

We establish a general bijective framework for encoding faces of some classical hyperplane arrangements. Precisely, we consider hyperplane arrangements in whose hyperplanes are all of the form for some and . Such an arrangement is \emph{strongly transitive} if it satisfies the following condition: if and for some and , then . For any strongly transitive arrangement , we establish a bijection between the faces of and some set of decorated plane trees.

Paper Structure

This paper contains 10 sections, 22 theorems, 87 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Notation and background
  3. Main results
  4. Examples
  5. From the Catalan arrangement to the Shi arrangement, and back
  6. Multi-Catalan arrangements
  7. Generating function for faces of symmetric transitive arrangements
  8. Generating functions of Catalan and semiorder arrangements
  9. Generating functions of arbitrary symmetric transitive arrangements
  10. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 2.9

If an arrangement $\mathcal{A}\subseteq\mathcal{A}_m^n$ is transitive, then $\Phi_\mathcal{A}$ is a bijection between the set $\mathop{\mathrm{\mathcal{T}_m^n}}\nolimits(\mathcal{A})$ of trees and the set $\mathcal{R}(\mathcal{A})$ of regions of $\mathcal{A}$.

Figures (10)

  • Figure 1: The braid, Catalan, Shi, semiorder, and Linial arrangements in dimension $n=3$. Here and later, the braid-type arrangements in dimension 3 are represented by drawing their intersection with the hyperplane $\{(x_1,x_2,x_3)\mid x_1+x_2+x_3=0\}$.
  • Figure 2: The order $\preceq_T$ for a binary tree $T$. The vertices of $T$ are ordered as follows: $a\mathop{\mathrm{\prec_T}}\nolimits b\mathop{\mathrm{\prec_T}}\nolimits c\mathop{\mathrm{\prec_T}}\nolimits \cdots \mathop{\mathrm{\prec_T}}\nolimits s$. Here the horizontal placement of vertices corresponds to their drift in the tree. Hence, in this representation, the relation $v\prec_T w$ can be thought as meaning that either "$v$ is strictly to the left of $w$" in the tree $T$ or "$v$ is at the vertical of $w$ but below $w$" in the tree $T$.
  • Figure 3: The bijection $\overline{\Phi}_\mathcal{A}$ for a strongly transitive arrangement $\mathcal{A}\subseteq \mathcal{A}_1^3$. The marked edges of the trees in $\mathop{\mathrm{\overline{\mathcal{T}}}}\nolimits_1^3(\mathcal{A})$ are indicated by bold lines.
  • Figure 4: (a) Decomposition of a marked tree in $\mathop{\mathrm{\overline{\mathcal{T}}}}\nolimits_m=\mathop{\mathrm{\overline{\mathcal{T}}}}\nolimits([-m;m])$ at the root-block. (b) A partition of the set $\mathop{\mathrm{\overline{\mathcal{T}}}}\nolimits_m'=\mathop{\mathrm{\overline{\mathcal{T}}}}\nolimits([-m;m]\setminus\{0\})$.
  • Figure 5: The bijection $\overline{\Phi}_\mathcal{A}$ for the semiorder arrangement $\mathcal{A}\subseteq \mathcal{A}_{\{-1,1\}}^3$.
  • ...and 5 more figures

Theorems & Definitions (60)

  • Definition 2.1
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.9: OB
  • Definition 3.1
  • Example 3.2
  • ...and 50 more