Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Labeling regions in deformations of graphical arrangements

Gábor Hetyei

TL;DR

This work develops a unified inequality-based framework for counting and labeling regions in deformations of graphical (braid) arrangements by encoding regions as weighted digraphs and applying Carver–Farkas duality together with the Flow Decomposition Theorem. It establishes a bijection between regions and valid $m$-acyclic weighted digraphs, with bounded regions corresponding to strongly connected digraphs, and introduces the gain poset to organize region structure across various deformation families. The framework recovers and extends key results for the Linial, Shi, extended Shi, Ish, and Fuss–Catalan arrangements, provides new labeling mechanisms (including for the $ ext{a}$-Catalan family), and connects to Athanasiadis–Linusson diagrams and ceiling diagrams, enabling efficient counting and structural insight across a broad class of hyperplane arrangements. The methods yield practical tools for proving injectivity/surjectivity of Pak–Stanley-type labelings, counting regions via generalized parking functions, and understanding when bounded regions arise through strong connectivity and gain-structure considerations.

Abstract

Combining Carver's variant of the Farkas' lemma with the Flow Decomposition Theorem we show that the regions of any deformation of a graphical arrangement may be bijectively labeled with a set of weighted digraphs containing directed cycles of negative weight only. Bounded regions correspond to strongly connected digraphs. The study of the resulting labelings allows us to add the omitted details in Stanley's proof on the injectivity of the Pak-Stanley labeling of the regions of the extended Shi arrangement, to generalize the ceiling diagrams in the deleted Shi and Ish arrangements studied by Armstrong and Rhoades and to introduce a new labeling of the regions in the Fuss-Catalan arrangement. We also point out that Athanasiadis-Linusson labelings may be used to directly count regions in a class of arrangements properly containing the extended Shi arrangement and the Fuss-Catalan arrangement.

Labeling regions in deformations of graphical arrangements

TL;DR

This work develops a unified inequality-based framework for counting and labeling regions in deformations of graphical (braid) arrangements by encoding regions as weighted digraphs and applying Carver–Farkas duality together with the Flow Decomposition Theorem. It establishes a bijection between regions and valid -acyclic weighted digraphs, with bounded regions corresponding to strongly connected digraphs, and introduces the gain poset to organize region structure across various deformation families. The framework recovers and extends key results for the Linial, Shi, extended Shi, Ish, and Fuss–Catalan arrangements, provides new labeling mechanisms (including for the -Catalan family), and connects to Athanasiadis–Linusson diagrams and ceiling diagrams, enabling efficient counting and structural insight across a broad class of hyperplane arrangements. The methods yield practical tools for proving injectivity/surjectivity of Pak–Stanley-type labelings, counting regions via generalized parking functions, and understanding when bounded regions arise through strong connectivity and gain-structure considerations.

Abstract

Combining Carver's variant of the Farkas' lemma with the Flow Decomposition Theorem we show that the regions of any deformation of a graphical arrangement may be bijectively labeled with a set of weighted digraphs containing directed cycles of negative weight only. Bounded regions correspond to strongly connected digraphs. The study of the resulting labelings allows us to add the omitted details in Stanley's proof on the injectivity of the Pak-Stanley labeling of the regions of the extended Shi arrangement, to generalize the ceiling diagrams in the deleted Shi and Ish arrangements studied by Armstrong and Rhoades and to introduce a new labeling of the regions in the Fuss-Catalan arrangement. We also point out that Athanasiadis-Linusson labelings may be used to directly count regions in a class of arrangements properly containing the extended Shi arrangement and the Fuss-Catalan arrangement.
Paper Structure (10 sections, 53 theorems, 83 equations, 8 figures, 1 table)

This paper contains 10 sections, 53 theorems, 83 equations, 8 figures, 1 table.

Table of Contents

  1. Preliminaries
  2. Two classical results
  3. Deformations of a graphical arrangement
  4. Labeling regions in deformations of graphical arrangements
  5. The poset of gains and sparse deformations
  6. Separated deformations and the weak triangle inequality
  7. Contiguous integral deformations
  8. Extended Shi arrangements
  9. The simplification lemma and ceiling hyperplanes
  10. The $a$-Catalan arrangements

Key Result

Lemma 1.1

Let $A$ be a real $m\times n$ matrix and let $b$ be a real $n\times 1$ column vector. Then the system of inequalities $Ax<b$ has no solution if and only if there is a nonzero real $m\times 1$ row vector $y$ satisfying $y\geq 0$, $yA=0$ and $yb\leq 0$.

Figures (8)

  • Figure 1: A shortest $m$-ascending cycle of length $5$
  • Figure 2: A valid weighted digraph with a minimal $m$-ascending $4$-cycle
  • Figure 3: An Athanasiadis-Linusson diagram
  • Figure 4: A rooted tree encoding an Athanasiadis-Linusson diagram
  • Figure 5: The weighted digraph associated to Example \ref{['ex:nonwall']}.
  • ...and 3 more figures

Theorems & Definitions (130)

  • Lemma 1.1: Carver-Farkas
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • proof
  • ...and 120 more