Parallel Translates of Represented Matroids

TL;DR

This paper investigates the structure of hyperplane arrangements derived from a represented matroid and its translates, classifying their intersection lattices and characteristic polynomials through associated derived arrangements.

Contribution

It introduces a classification of parallel translates of hyperplane arrangements associated with a matroid using the derived arrangement and provides decomposition formulas for their characteristic polynomials.

Findings

Classification of intersection semi-lattices and characteristic polynomials for all translates.

Establishment of a connection between the arrangement of translates and the derived arrangement.

Decomposition formulas for characteristic polynomials of translated arrangements.

Abstract

Given an $\Bbb{F}$-represented matroid $(M,\rho)$ with the ground set $[m]$, the representation $\rho$ naturally defines a hyperplane arrangement $\mathcal{A}_\rho$. We will study its parallel translates $\mathcal{A}_{\rho,{g}}$ of $\mathcal{A}_\rho$ for all ${ g}\in \mathbb{F}^m$. Its intersection semi-lattices $L(\mathcal{A}_{\rho,{ g}})$ and the characteristic polynomials $\chi(\mathcal{A}_{\rho,{ g}},t)$ will be classified by the intersection lattice of the derived arrangement $\mathcal{A}_{\delta\rho}$, which is a hyperplane arrangement associated with the derived matroid $(\delta M,\delta\rho)$ and also known as the discriminantal arrangement in the literature. As a byproduct, we obtain a comparison result and a decomposition formula on the characteristic polynomials $\chi(\mathcal{A}_{\rho,{ g}},t)$.