Peterson varieties and toric orbifolds associated to Cartan matrices

TL;DR

This paper establishes a direct algebraic connection between Peterson varieties and toric orbifolds associated with root systems by constructing an explicit morphism that induces an isomorphism of their rational cohomology rings.

Contribution

It provides the first explicit morphism linking Peterson varieties to toric orbifolds, demonstrating their cohomological equivalence beyond known algebraic coincidences.

Findings

Constructed an explicit morphism between Peterson varieties and toric orbifolds.

Proved the morphism induces an isomorphism of rational cohomology rings.

Confirmed the algebraic coincidence is not accidental.

Abstract

The Peterson variety is a remarkable variety introduced by Dale Peterson to describe the quantum cohomology rings of all the partial flag varieties. The rational cohomology ring of the Peterson variety is known to be isomorphic to that of a particular toric orbifold which naturally arises from the given root system. In this paper, we show that it is not an accidental algebraic coincidence; we construct an explicit morphism from the Peterson variety to the toric orbifold which induces a ring isomorphism between their rational cohomology rings.