{
  "id": "1410.7419",
  "version": "v1",
  "published": "2014-10-08T19:46:56.000Z",
  "updated": "2014-10-08T19:46:56.000Z",
  "title": "Cohomology classes of rank varieties and a conjecture of Liu",
  "authors": [
    "Brendan Pawlowski"
  ],
  "comment": "16 pages",
  "categories": [
    "math.CO",
    "math.AG"
  ],
  "abstract": "To each finite subset of a discrete grid $\\mathbb{N} \\times \\mathbb{N}$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group (a Specht module). Liu has conjectured that the cohomology class of a diagram variety is represented by the Frobenius characteristic of the corresponding Specht module. We give a counterexample to this conjecture. However, we show that for the diagram variety of a permutation diagram, Liu's conjectured cohomology class $\\sigma$ is at least an upper bound on the actual class $\\tau$, in the sense that $\\sigma - \\tau$ is a nonnegative linear combination of Schubert classes. To do this, we consider a degeneration of Coskun's rank varieties which contains the appropriate diagram variety as a component. Rank varieties are instances of Knutson-Lam-Speyer's positroid varieties, whose cohomology classes are represented by affine Stanley symmetric functions. We show that the cohomology class of a rank variety is in fact represented by an ordinary Stanley symmetric function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-08T19:46:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "05E10",
      "14N15"
    ],
    "keywords": [
      "rank variety",
      "cohomology classes",
      "ordinary stanley symmetric function",
      "specht module",
      "affine stanley symmetric functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.7419P"
    }
  }
}