Supersolvability and Freeness for $\psi$-graphical Arrangements

TL;DR

This paper proves a conjecture characterizing when $ ext{psi}$-graphical arrangements are supersolvable, extending known results from graphical arrangements, and discusses conditions for their freeness.

Contribution

It proves the conjecture that $ ext{psi}$-graphical arrangements are supersolvable if and only if certain graph conditions hold, extending the theory of graphical arrangements.

Findings

Proved the conjecture relating supersolvability to graph properties.

Established conditions under which $ ext{psi}$-graphical arrangements are free.

Extended the characterization from graphical to $ ext{psi}$-graphical arrangements.

Abstract

Let $G$ be a simple graph on the vertex set $\{v_1,\dots,v_n\}$ with edge set $E$. Let $K$ be a field. The graphical arrangement $\mathcal{A}_G$ in $K^n$ is the arrangement $x_i-x_j=0, v_iv_j \in E$. An arrangement $\mathcal{A}$ is supersolvable if the intersection lattice $L(c(\mathcal{A}))$ of the cone $c(\mathcal{A})$ contains a maximal chain of modular elements. The second author has shown that a graphical arrangement $\mathcal{A}_G$ is supersolvable if and only if $G$ is a chordal graph. He later considered a generalization of graphical arrangements which are called $\psi$-graphical arrangements. He conjectured a characterization of the supersolvability and freeness (in the sense of Terao) of a $\psi$-graphical arrangement. We provide a proof of the first conjecture and state some conditions on free $\psi$-graphical arrangements.