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Separator-based derivations of graphic arrangements

Leonie Mühlherr

TL;DR

The paper addresses constructing the module of logarithmic derivations $D(\mathcal{A}(G))$ for graphic hyperplane arrangements by introducing separator-based derivations, thereby generalizing the braid-arrangement (complete graph) basis to arbitrary graphs. It proves a main result: a minimal generating set for $D(\mathcal{A}(G))$ exists consisting of $\theta_0,\dots,\theta_\kappa$ (where $\kappa=\kappa(G)$) together with a finite set of separator-based $\mathcal{A}$-derivations, providing a constructive method to obtain a basis. The work yields concrete bounds on the derivation degree sequence, namely $d$ satisfies $\max\{c-1,t_{\max}\}\le d \le \Delta(G)$, linking combinatorial graph properties (clique number, separator size, maximum degree) to algebraic structure. This framework offers a new avenue for analyzing non-free graphic arrangements and suggests directions for extending the approach to MAT-free, flag-accurate, and related classes of hyperplane arrangements.

Abstract

The class of subarrangements of the well-studied braid arrangement, so-called graphic hyperplane arrangements, is important for analysing new concepts and properties in hyperplane arrangement theory. While there is a nice characterization of free graphic arrangements, many interesting questions beyond the free case remain open. This paper introduces an explicit set of generators for the module of logarithmic derivations of a general graphic arrangement based on graph separators. We obtain new insights in the derivation degree sequence in the non-free case and give bounds on the highest degree in the sequence. Moreover, this can broadly be used for future research in this area.

Separator-based derivations of graphic arrangements

TL;DR

The paper addresses constructing the module of logarithmic derivations for graphic hyperplane arrangements by introducing separator-based derivations, thereby generalizing the braid-arrangement (complete graph) basis to arbitrary graphs. It proves a main result: a minimal generating set for exists consisting of (where ) together with a finite set of separator-based -derivations, providing a constructive method to obtain a basis. The work yields concrete bounds on the derivation degree sequence, namely satisfies , linking combinatorial graph properties (clique number, separator size, maximum degree) to algebraic structure. This framework offers a new avenue for analyzing non-free graphic arrangements and suggests directions for extending the approach to MAT-free, flag-accurate, and related classes of hyperplane arrangements.

Abstract

The class of subarrangements of the well-studied braid arrangement, so-called graphic hyperplane arrangements, is important for analysing new concepts and properties in hyperplane arrangement theory. While there is a nice characterization of free graphic arrangements, many interesting questions beyond the free case remain open. This paper introduces an explicit set of generators for the module of logarithmic derivations of a general graphic arrangement based on graph separators. We obtain new insights in the derivation degree sequence in the non-free case and give bounds on the highest degree in the sequence. Moreover, this can broadly be used for future research in this area.
Paper Structure (6 sections, 20 theorems, 42 equations, 7 figures)

This paper contains 6 sections, 20 theorems, 42 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph Theory
  4. Hyperplane Arrangements
  5. Constructing a Generating Set
  6. On minimal generating sets

Key Result

Theorem 1

Define for $k \in \mathbb{N}_0$ the derivation $\theta_k = \sum_{i =1}^\ell x_i^k \cdot \partial_{x_i}$, then $\theta_0,\dots, \theta_{\ell-1}$ is a basis of the module $D(\mathcal{A}(K_\ell))$.

Figures (7)

  • Figure 1: Example graph with minimal separators of different sizes.
  • Figure 2: Example of a chordal graph with 4-dimensional $\mathcal{A}(G)$-derivation module
  • Figure 3: The graph $G$ with its unique separator poset.
  • Figure 4: The graph $G$ with a generated $\mathscr{P}(G)$-poset. The elements that are generated by $\mathscr{P}(G)$ are in italic. Note that $\mathscr{P}(G)$ is complete.
  • Figure 5: The 6-antihole and its separator poset. For all antiholes it holds that the minimal separators of $\overline{C_\ell}$ are always the neighbourhoods of the vertices, thus of cardinality $\ell-3$, separating them from their neighbours in the cycle graph. The separator-posets $\mathscr{P}({\overline{C_\ell}})$ are thus complete for all $\ell$.
  • ...and 2 more figures

Theorems & Definitions (68)

  • Definition 1.1
  • Theorem : Saito
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Proposition 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • ...and 58 more