{
  "id": "1707.02178",
  "version": "v1",
  "published": "2017-07-07T13:56:17.000Z",
  "updated": "2017-07-07T13:56:17.000Z",
  "title": "Applying Parabolic Peterson: Affine Algebras and the Quantum Cohomology of the Grassmannian",
  "authors": [
    "Jonathan Cookmeyer",
    "Elizabeth Milićević"
  ],
  "comment": "38 pages, most figures best viewed in color",
  "categories": [
    "math.CO",
    "math.AG"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-07T13:56:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M15",
      "05E05",
      "20F55",
      "14N15",
      "14N35"
    ],
    "keywords": [
      "quantum cohomology",
      "applying parabolic peterson",
      "affine algebras",
      "affine grassmannian",
      "affine niltemperley-lieb algebra arises"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}