Cohomology of the toric arrangement associated with $A_n$

TL;DR

This paper computes the total cohomology of the complement of a specific toric arrangement related to the root system $A_n$, revealing its structure as a Weyl group representation and providing explicit formulas for its Poincaré polynomial.

Contribution

It introduces a method to compute the cohomology of the arrangement complement and offers explicit formulas for the Poincaré polynomial, advancing understanding of toric arrangements associated with $A_n$.

Findings

Total cohomology computed as a Weyl group representation

Explicit formula for the Poincaré polynomial derived

Provides multiple proofs for the Poincaré polynomial formula

Abstract

We compute the total cohomology of the complement of the toric arrangement associated to the root system $A_n$ as a representation of the corresponding Weyl group via fixed point theory of a "twisted" action of the group. We also provide several proofs of an explicit formula for the Poincar\'e polynomial of the complement of the toric arrangement associated with $A_n$.