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arXiv · 1707.02178

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

Tessa Cookmeyer·Elizabeth Milićević
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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.

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Tessa Cookmeyer, Elizabeth Milićević. 2017-07-07. Applying Parabolic Peterson: Affine Algebras and the Quantum Cohomology of the Grassmannian. https://arxiv.org/abs/1707.02178

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