Parallel Translates of Represented Matroids

Given an $\mathbb{F}$-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 $\chi ({\mathcal{A}}_{\rho ,g},t)$.