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

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.

Keywords

Peterson isomorphism, quantum cohomology, non-commutative $k$-Schur functions, Grassmannian, affine nilTemperley–Lieb algebra, Schubert calculus

2010 Mathematics Subject Classification

Primary 05E05, 14M15. Secondary 14N15, 14N35, 20F55.

Full Text (PDF format)

Elizabeth Milićević was partially supported by the Max-Planck-Institut für Mathematik and NSF grant DMS–1600982.

Received 6 August 2017

Published 7 December 2018