One-element Extensions of Hyperplane Arrangements

TL;DR

This paper classifies one-element extensions of hyperplane arrangements using the induced adjoint arrangement and explores how various combinatorial invariants behave on these strata, providing new convolution formulas for characteristic polynomials.

Contribution

It introduces a classification of one-element extensions via the induced adjoint arrangement and shows invariance and order-preservation of key combinatorial invariants on associated strata.

Findings

Invariants are constant on strata associated with the induced adjoint arrangement.

Invariants are order-preserving with respect to the intersection lattice.

Provides a convolution formula for characteristic polynomials over finite fields.

Abstract

We classify one-element extensions of a hyperplane arrangement by the induced adjoint arrangement. Based on the classification, several kinds of combinatorial invariants including Whitney polynomials, characteristic polynomials, Whitney numbers and face numbers, are constants on those strata associated with the induced adjoint arrangement, and also order-preserving with respect to the intersection lattice of the induced adjoint arrangement. As a byproduct, we obtain a convolution formula on the characteristic polynomials $\chi(\mathcal{A}+H_{\bm\alpha,a},t)$ when $\mathcal{A}$ is defined over a finite field $\mathbb{F}_q$ or a rational arrangement.