{
  "id": "1804.03701",
  "version": "v1",
  "published": "2018-04-10T19:59:34.000Z",
  "updated": "2018-04-10T19:59:34.000Z",
  "title": "Catalan functions and $k$-Schur positivity",
  "authors": [
    "Jonah Blasiak",
    "Jennifer Morse",
    "Anna Pun",
    "Daniel Summers"
  ],
  "comment": "43 pages, 2 figures",
  "categories": [
    "math.CO",
    "math.QA"
  ],
  "abstract": "We prove that graded $k$-Schur functions are $G$-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We expose a new miraculous shift invariance property of the graded $k$-Schur functions and resolve the Schur positivity and $k$-branching conjectures in the strongest possible terms by providing direct combinatorial formulas using strong marked tableaux.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-10T19:59:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "14N15",
      "05A30",
      "33D52"
    ],
    "keywords": [
      "schur positivity",
      "catalan functions",
      "schur functions",
      "direct combinatorial formulas",
      "miraculous shift invariance property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}