Characterizing Nice Partition of Graphical Arrangements

TL;DR

This work investigates the interplay between graph structure and niceness in graphical hyperplane arrangements. It proves that for graphical arrangements, the properties of supersolvability, freeness, and the existence of nice partitions are all captured by chordal graphs, and that every nice partition of a graphical arrangement $ abla$ arises from a maximal modular chain in its intersection lattice $L( abla)$, with constructive methods to obtain the chain. It also provides two converses to classical Orlik–Terao results: (i) a partition is nice precisely when the localized characteristic polynomials satisfy $\\chi(\\mathcal{A}_X,t)=t^{n-l}\\prod_{i=1}^{l}(t-|\\\pi_i\\cap \\mathcal{A}_X|)$ for all $X$, and (ii) a nice partition induced by a maximal chain implies the chain is modular, hence the arrangement is supersolvable. Together, these results unify the criteria for niceness, modular chains, and supersolvability in graphical arrangements and supply explicit constructions from block-decomposed graphs.

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.

Figures (4)

• Figure 1: The graph $G$ in Example \ref{['ex']}, which consists of two blocks
• Figure 2: The illustration of Lemma \ref{['mainprop1']}
• Figure 3: Illustration of Example \ref{['ex:mc']}
• Figure 4: The second illustration of Example \ref{['ex:mc']}

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1
• Proposition 2.2
• Lemma 3.1
• proof
• Example 3.2
• proof : Proof of Theorem \ref{['chordal-nice']}