Cohomology of Complements of Toric Arrangements Associated to Root Systems

TL;DR

This paper develops an algorithm to compute the cohomology of complements of toric arrangements linked to root systems, providing explicit results for exceptional types and formulas for classical types.

Contribution

It introduces a novel algorithm for cohomology computation of toric arrangements associated with root systems, including explicit calculations for exceptional types and a Poincaré polynomial formula for type A.

Findings

Computed cohomology for exceptional root systems G2, F4, E6, E7

Derived a formula for the Poincaré polynomial of A_n arrangements

Provided a general algorithm for cohomology calculations

Abstract

We compute the cohomology of the complement of toric arrangements associated to root systems as representations of the corresponding Weyl groups. Specifically, we develop an algorithm for computing the cohomology of the complement of toric arrangements associated to general root systems and we carry out this computation for the exceptional root systems $G_2$, $F_4$, $E_6$ and $E_7$. We also compute the total cohomology of the complement of the toric arrangement associated to $A_n$ as a representation of the Weyl group and give a formula for its Poincar\'e polynomial.