Region level via centralization for hyperplane arrangements and beyond

TL;DR

The paper refines region enumeration in real hyperplane arrangements by introducing centralization to count regions by level, producing a bijective proof that shows $r_\ell(\mathcal{A})$ is determined by the intersection poset $\mathcal{L}(\mathcal{A})$ and extends the framework to geometric semilattices. It derives a general expression for the characteristic polynomial $\chi_M(t)$ of a geometric semilattice and proves a key decomposition $\chi_M(t) = \sum_{S\in\underline{M}} \chi_{\underline{M}^S}(t) \chi_{M_S}(1)$, enabling level-wise counting and invariance results. The work unifies and extends prior results on level counting, providing explicit formulas for deformations of the braid arrangement (exponential arrangements) and revealing binomial-type generating identities for the associated level counts. These results offer new combinatorial tools for both realizable and nonrealizable semilattices, with implications for generating functions and characteristic polynomials in broader settings. Overall, the paper bridges geometric, combinatorial, and algebraic perspectives to deepen understanding of region structure in hyperplane arrangements and their deformations.

Abstract

In "Faces of a Hyperplane Arrangement Enumerated by Ideal Dimension, with Applications to Plane, Plaids, and Shi," Zaslavsky showed how to compute the number $r_\ell(\mathcal{A})$ of regions of a real hyperplane arrangement $\mathcal{A}$ with a given level, refining his well known enumeration of regions and relatively bounded regions. We restate this theorem in terms of a construction called the centralization of $\mathcal{A}$, give a bijective proof, and then apply it in two ways to answer questions concerning the concept of level. Firstly, a consequence of this enumeration is that $r_\ell(\mathcal{A})$ depends only on the intersection poset $\mathcal{L}(\mathcal{A})$, such that both $r_\ell$ and centralization can be defined in the more general setting of geometric semilattices. In this context we derive a very general expression for the characteristic polynomial of a geometric semilattice with several interesting corollaries. Secondly, recent investigations into the phenomenon of level have made little use of Zaslavsky's level-counting theorem, but it can be applied to obtain or generalize many of their results. In particular we show how exponential generating function identities (arXiv:2410.10198, arXiv:2411.02971) and an expression giving the characteristic polynomial in terms of $r_\ell$ (arXiv:2411.03756) can be derived for deformations of the braid arrangement.

Figures (3)

• Figure 1: A pair of arrangements with the same characteristic polynomial, but different values of $r_1$ and $r_2$.
• Figure 2: The cone of the lattice of flats of the matroid $U_{2,4}$, minus its maximum element. A copy of the original semilattice is in blue, with its centralization 'above' it in red.
• Figure 3: $\text{Shi}(3) = \{x_i - x_j = 0,1 : 1 \leq i < j \leq 3\}$

Theorems & Definitions (52)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• proof
• Theorem 1.4
• Theorem 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5