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

TL;DR

This paper introduces a digraph-based framework for hyperplane arrangements that generalizes several classical arrangements and proves their freeness, including new results on arrangements interpolating between Shi and Ish.

Contribution

It defines digraph operations that preserve arrangement properties and establishes the freeness of arrangements between Shi and Ish, extending prior results in the field.

Findings

Arrangements between Shi and Ish all have the same characteristic polynomial with integer roots.

The introduced operations preserve characteristic polynomials and freeness.

Only the Ish arrangement among these has a supersolvable cone.

Abstract

We introduce and study a digraph analogue of Stanley's $\psi$-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 $\psi$, 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.