{
  "id": "1008.4302",
  "version": "v1",
  "published": "2010-08-25T15:09:49.000Z",
  "updated": "2010-08-25T15:09:49.000Z",
  "title": "Puzzles, positroid varieties, and equivariant K-theory of Grassmannians",
  "authors": [
    "Allen Knutson"
  ],
  "comment": "30 pages; color helpful but not essential",
  "categories": [
    "math.AG",
    "math.CO",
    "math.KT"
  ],
  "abstract": "Vakil studied the intersection theory of Schubert varieties in the Grassmannian in a very direct way: he degenerated the intersection of a Schubert variety X_mu and opposite Schubert variety X^nu to a union {X^lambda}, with repetition. This degeneration proceeds in stages, and along the way he met a collection of more complicated subvarieties, which he identified as the closures of certain locally closed sets. We show that Vakil's varieties are _positroid varieties_, which in particular shows they are normal, Cohen-Macaulay, have rational singularities, and are defined by the vanishing of Pl\\\"ucker coordinates [Knutson-Lam-Speyer]. We determine the equations of the Vakil variety associated to a partially filled ``puzzle'' (building on the appendix to [Vakil]), and extend Vakil's proof to give a geometric proof of the puzzle rule from [Knutson-Tao '03] for equivariant Schubert calculus. The recent paper [Anderson-Griffeth-Miller] establishes (abstractly; without a formula) three positivity results in equivariant K-theory of flag manifolds G/P. We demonstrate one of these concretely, giving a corresponding puzzle rule.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-08-25T15:09:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M15",
      "05E99"
    ],
    "keywords": [
      "equivariant k-theory",
      "positroid varieties",
      "grassmannian",
      "puzzle rule",
      "flag manifolds g/p"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.4302K"
    }
  }
}