Combinatorics and topology of toric arrangements defined by root systems

TL;DR

This paper studies the combinatorial and topological properties of toric arrangements derived from root systems, providing explicit formulas for layer counts, and computing topological invariants like Euler characteristic and Poincaré polynomial.

Contribution

It introduces a method to analyze the layers of toric arrangements from root systems and derives explicit formulas for their enumeration and topological invariants.

Findings

Explicit formula for counting layers using affine Dynkin diagrams

Reduction technique to zero-dimensional layers

Computed Euler characteristic and Poincaré polynomial of the arrangement complement

Abstract

Given the toric (or toral) arrangement defined by a root system $\Phi$, we describe the poset of its layers (connected components of intersections) and we count its elements. Indeed we show how to reduce to zero-dimensional layers, and in this case we provide an explicit formula involving the maximal subdiagrams of the affine Dynkin diagram of $\Phi$. Then we compute the Euler characteristic and the Poincare' polynomial of the complement of the arrangement, which is the set of regular points of the torus.