Applying parabolic Peterson: affine algebras and the quantum cohomology of the Grassmannian

Document Type

Journal Article

Role

Author

Publisher

International Press

Journal Title

Journal of Combinatorics

Volume

10

Issue

1

First Page

129

Last Page

162

Publication Date

12-7-2018

Abstract

The Peterson isomorphism relates the homology of the affine Grass- mannian 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 iso- morphically 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 Pe- terson by showing that the affine nilTemperley-Lieb algebra arises naturally when forming the requisite quotient of the homology of the affine Grassmannian.

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