{
  "id": "1811.02490",
  "version": "v1",
  "published": "2018-11-06T17:02:56.000Z",
  "updated": "2018-11-06T17:02:56.000Z",
  "title": "$k$-Schur expansions of Catalan functions",
  "authors": [
    "Jonah Blasiak",
    "Jennifer Morse",
    "Anna Pun",
    "Daniel Summers"
  ],
  "comment": "33 pages",
  "categories": [
    "math.CO",
    "math.AG",
    "math.QA"
  ],
  "abstract": "We make a broad conjecture about the $k$-Schur positivity of Catalan functions, symmetric functions which generalize the (parabolic) Hall-Littlewood polynomials. We resolve the conjecture with positive combinatorial formulas in cases which address the $k$-Schur expansion of (1) Hall-Littlewood polynomials, proving the $q=0$ case of the strengthened Macdonald positivity conjecture of Lapointe, Lascoux, and Morse; (2) the product of a Schur function and a $k$-Schur function when the indexing partitions concatenate to a partition, describing a class of Gromov-Witten invariants for the quantum cohomology of complete flag varieties; (3) $k$-split polynomials, proving a substantial case of a problem of Broer and Shimozono-Weyman on parabolic Hall-Littlewood polynomials. In addition, we prove the conjecture that $k$-Schur functions defined in terms of $k$-split polynomials agree with strong tableau $k$-Schur functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-06T17:02:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "14N15",
      "14N35",
      "05A30",
      "33D52"
    ],
    "keywords": [
      "catalan functions",
      "schur expansion",
      "schur function",
      "strengthened macdonald positivity conjecture",
      "complete flag varieties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}