Parallel translates of represented matroids

Given an F-represented matroid (M,ρ) with the ground set [m], 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 characteristic polynomials χ(Aρ,g,t) be classified by lattice derived Aδρ, which is associated (δM,δρ) also known as discriminantal in literature. As byproduct, we obtain comparison result decomposition formula on χ(Aρ,g,t).