Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Vertex-weighted Digraphs and Freeness of Arrangements Between Shi and Ish

Takuro Abe, Tan Nhat Tran, Shuhei Tsujie

Abstract

We introduce and study a digraph analogue of Stanley's $ψ$-graphical arrangements from the perspectives of combinatorics and freeness. Our arrangements form a common generalization of various classes of arrangements in literature including the Catalan arrangement, the Shi arrangement, the Ish arrangement, and especially the arrangements interpolating between Shi and Ish recently introduced by Duarte and Guedes de Oliveira. The arrangements between Shi and Ish all are proved to have the same characteristic polynomial with all integer roots, thus raising the natural question of their freeness. We define two operations on digraphs, which we shall call king and coking elimination operations and prove that subject to certain conditions on the weight $ψ$, the operations preserve the characteristic polynomials and freeness of the associated arrangements. As an application, we affirmatively prove that the arrangements between Shi and Ish all are free, and among them only the Ish arrangement has supersolvable cone.

Vertex-weighted Digraphs and Freeness of Arrangements Between Shi and Ish

Abstract

We introduce and study a digraph analogue of Stanley's -graphical arrangements from the perspectives of combinatorics and freeness. Our arrangements form a common generalization of various classes of arrangements in literature including the Catalan arrangement, the Shi arrangement, the Ish arrangement, and especially the arrangements interpolating between Shi and Ish recently introduced by Duarte and Guedes de Oliveira. The arrangements between Shi and Ish all are proved to have the same characteristic polynomial with all integer roots, thus raising the natural question of their freeness. We define two operations on digraphs, which we shall call king and coking elimination operations and prove that subject to certain conditions on the weight , the operations preserve the characteristic polynomials and freeness of the associated arrangements. As an application, we affirmatively prove that the arrangements between Shi and Ish all are free, and among them only the Ish arrangement has supersolvable cone.

Paper Structure

This paper contains 7 sections, 34 theorems, 66 equations, 4 figures.

Table of Contents

  1. Introduction
  2. $\psi$-digraphical arrangements
  3. Stability of characteristic polynomials
  4. Stability of freeness
  5. Freeness of Ziegler restrictions
  6. Local freeness
  7. Proof of Theorem \ref{['thm:freeness-stable']} and applications

Key Result

Theorem 1.2

If $\mathcal{A}$ is free with $\exp(\mathcal{A}) =\{d_{1}, \dots, d_{\ell}\}$, then

Figures (4)

  • Figure 1: From left to right: the transitive tournament, the complete digraph, and the edgeless digraph on $4$ vertices.
  • Figure 2: $(k,5)$-Shi-Ish arrangements $\mathcal{A}_{5}^{k}$$(2 \le k \le 5)$. Here a vertex is in bold symbol placed next to its weight.
  • Figure 3: Characterization for freeness of the arrangements of type (L\ref{['localization type 2']}).
  • Figure 4: A sequence of CEOs and KEOs applying to the Catalan arrangement $\mathop{\mathrm{Cat}}\nolimits(4)$.

Theorems & Definitions (68)

  • Definition 1.1
  • Theorem 1.2: Factorization Theorem, e.g., T81, OT92
  • Theorem 1.3: e.g., Headley97A96Arms13AR12
  • Theorem 1.4: Atha98AST17Yo04
  • Theorem 1.5: DG18
  • Theorem 1.6
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Example 2.4
  • ...and 58 more