Combinatorics and topology of toric arrangements defined by root systems

Abstract

Given the toric (or toral) arrangement defined by a root system $\Phi$, 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 $\Phi$. Then we compute the Euler characteristic and the Poincar\'{e} polynomial of the complement of the arrangement, that is the set of regular points of the torus.