Region level via centralization for hyperplane arrangements and beyond

TL;DR

This paper revisits Zaslavsky's enumeration of hyperplane arrangement regions by level, introduces a centralization construction, and extends the results to geometric semilattices and arrangement deformations.

Contribution

It restates Zaslavsky's theorem using centralization, generalizes the concept to geometric semilattices, and applies it to deformations of the braid arrangement.

Findings

Enumeration of regions depends only on the intersection poset.

Derived a general expression for the characteristic polynomial of geometric semilattices.

Applied the theorem to obtain identities and formulas for arrangement 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.