Applying Parabolic Peterson: Affine Algebras and the Quantum Cohomology of the Grassmannian

TL;DR

This paper explores the parabolic Peterson isomorphism relating affine Grassmannian homology to quantum cohomology of Grassmannians, providing explicit methods for quantum Schubert calculus and connecting to affine algebra structures.

Contribution

It offers a detailed exposition of the parabolic Peterson isomorphism for Grassmannians, including explicit computational recipes and algebraic interpretations.

Findings

Explicit recipe for quantum Schubert calculus using non-commutative k-Schur functions

Recasting Postnikov's approach via the affine nilTemperley-Lieb algebra

Establishing the isomorphism between affine algebra quotients and quantum cohomology

Abstract

The Peterson isomorphism relates the homology of the affine Grassmannian to the quantum cohomology of any flag variety. In the case of a partial flag, Peterson's map is only a surjection, and one needs to quotient by a suitable ideal on the affine side to map isomorphically onto the quantum cohomology. We provide a detailed exposition of this parabolic Peterson isomorphism in the case of the Grassmannian of m-planes in complex n-space, including an explicit recipe for doing quantum Schubert calculus in terms of the appropriate subset of non-commutative k-Schur functions. As an application, we recast Postnikov's affine approach to the quantum cohomology of the Grassmannian as a consequence of parabolic Peterson by showing that the affine nilTemperley-Lieb algebra arises naturally when forming the requisite quotient of the homology of the affine Grassmannian.