Characterizing Nice Partition of Graphical Arrangements

TL;DR

This paper explores the relationship between nice partitions and modular chains in graphical arrangements, establishing new characterizations and converses related to chordal graphs and intersection lattices.

Contribution

It proves that every nice partition corresponds to a maximal modular chain and provides two new converses to classical results on nice partitions.

Findings

Nice partitions correspond to maximal modular chains in the intersection lattice.

Established two converses to classical results of Orlik and Terao.

Characterized nice partitions explicitly for graphical arrangements.

Abstract

The successive works of Terao as well as Stanley revealed that, for graphical arrangements, supersolvability and the existence of nice partitions are equivalent properties, both characterized by chordal graphs. In this paper, we further prove that every nice partition of a graphical arrangement arises precisely from a maximal modular chain in its intersection lattice. Moreover, we establish two converses to classical results of Orlik and Terao on nice partitions.