Parallel translates of represented matroids

Abstract

Given an F-represented matroid (M,ρ) with the ground set [m], the representation ρ naturally defines a hyperplane arrangement Aρ. We will study its parallel translates Aρ,g of Aρ for all g∈Fm. Its intersection semi-lattices L(Aρ,g) and the characteristic polynomials χ(Aρ,g,t) will be classified by the intersection lattice of the derived arrangement Aδρ, which is a hyperplane arrangement associated with the derived matroid (δM,δρ) 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 χ(Aρ,g,t).