Journal of Combinatorics
Volume 10 (2019)
Applying parabolic Peterson: affine algebras and the quantum cohomology of the Grassmannian
Pages: 129 – 162
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.
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.
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