The equivariant cohomology rings of Peterson varieties in all Lie types

TL;DR

This paper provides a uniform, efficient presentation of the equivariant cohomology rings of Peterson varieties across all Lie types, extending previous results limited to type A.

Contribution

It introduces a quadratic ideal presentation of the cohomology ring for Peterson varieties in all Lie types, generalizing earlier type A results using Cartan matrices.

Findings

Quadratic ideal J describes the cohomology ring

Uniform presentation across all Lie types

Extension of previous type A results

Abstract

Let G be a complex semisimple linear algebraic group and let Pet be the associated Peterson variety in the flag variety G/B. The main theorem of this note gives an efficient presentation of the equivariant cohomology ring H^*_S(Pet) of the Peterson variety as a quotient of a polynomial ring by an ideal J generated by quadratic polynomials, in the spirit of the Borel presentation of the cohomology of the flag variety. Here the group S \cong \mathbb{C}^* is a certain subgroup of a maximal torus T of G. Our description of the ideal J uses the Cartan matrix and is uniform across Lie types. In our arguments we use the Monk formula and Giambelli formula for the equivariant cohomology rings of Peterson varieties for all Lie types, as obtained in the work of Drellich. Our result generalizes a previous theorem of Fukukawa-Harada-Masuda, which was only for Lie type A.