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

Cookmeyer, Jonathan; Milićević, Elizabeth

doi:10.4310/JOC.2019.v10.n1.a6

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

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Milićević, Elizabeth
Max Planck Institute for Mathematics, Max Planck Society;

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http://dx.doi.org/10.4310/JOC.2019.v10.n1.a6
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Citation

Cookmeyer, J., & Milićević, E. (2019). Applying parabolic Peterson: affine algebras and the quantum cohomology of the Grassmannian. Journal of Combinatorics, 10(1), 129-162. doi:10.4310/JOC.2019.v10.n1.a6.

Cite as: https://hdl.handle.net/21.11116/0000-0003-D8FC-E

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.