{
  "id": "2007.04952",
  "version": "v1",
  "published": "2020-07-09T17:27:42.000Z",
  "updated": "2020-07-09T17:27:42.000Z",
  "title": "Demazure crystals and the Schur positivity of Catalan functions",
  "authors": [
    "Jonah Blasiak",
    "Jennifer Morse",
    "Anna Pun"
  ],
  "comment": "52 pages, 5 figures",
  "categories": [
    "math.CO",
    "math.QA"
  ],
  "abstract": "Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety, are a rich class of symmetric functions which include $k$-Schur functions and parabolic Hall-Littlewood polynomials. We prove that Catalan functions indexed by partition weight are the characters of $U_q(\\widehat{\\mathfrak{sl}}_\\ell)$-generalized Demazure crystals as studied by Lakshmibai-Littelmann-Magyar and Naoi. We obtain Schur positive formulas for these functions, settling conjectures of Chen-Haiman and Shimozono-Weyman. Our approach more generally gives key positive formulas for graded Euler characteristics of certain vector bundles on Schubert varieties by matching them to characters of generalized Demazure crystals.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-09T17:27:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "05E05",
      "81R50",
      "33D52",
      "14M15"
    ],
    "keywords": [
      "catalan functions",
      "schur positivity",
      "graded euler characteristics",
      "generalized demazure crystals",
      "vector bundles"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}