On arrangements of hyperplanes from connected subgraphs

TL;DR

This paper studies hyperplane arrangements derived from connected subgraphs of a fixed graph, characterizing their algebraic and combinatorial properties such as freeness, simpliciality, and supersolvability.

Contribution

It provides a complete characterization of when these arrangements are free, based on the structure of the underlying graph.

Findings

Arrangements are free if and only if the graph is a cycle, path, almost path, or a path with a triangle.

Includes the resonance arrangement and certain Weyl subarrangements.

Characterizes arrangements as simplicial, factored, or supersolvable based on graph structure.

Abstract

We investigate arrangements of hyperplanes whose normal vectors are given by connected subgraphs of a fixed graph. These include the resonance arrangement and certain ideal subarrangements of Weyl arrangements. We characterize those which are free, simplicial, factored, or supersolvable. In particular, such an arrangement is free if and only if the graph is a cycle, a path, an almost path, or a path with a triangle attached to it.