Combinatorics and topology of toric arrangements defined by root systems

Abstract

Given the toric (or toral) arrangement defined by a root system φ, we classify and count its components of each dimension. We show how to reduce to the case of 0-dimensional components, and in this case we give an explicit formula involving the maximal subdiagrams of the affine Dynkin diagram of φ. Then we compute the Euler characteristic and the Poincaré polynomial of the complement of the arrangement, which is the set of regular points of the torus.