{
  "id": "1807.03294",
  "version": "v1",
  "published": "2018-07-09T17:58:12.000Z",
  "updated": "2018-07-09T17:58:12.000Z",
  "title": "Crystal structures for symmetric Grothendieck polynomials",
  "authors": [
    "Cara Monical",
    "Oliver Pechenik",
    "Travis Scrimshaw"
  ],
  "comment": "44 pages, 6 figures",
  "categories": [
    "math.CO",
    "math.AG",
    "math.KT"
  ],
  "abstract": "The symmetric Grothendieck polynomials representing Schubert classes in the $K$-theory of Grassmannians are generating functions for semistandard set-valued tableaux. We construct a type $A_n$ crystal structure on these tableaux. This crystal yields a new combinatorial formula for decomposing symmetric Grothendieck polynomials into Schur polynomials. For single-columns and single-rows, we give a new combinatorial interpretation of Lascoux polynomials (K-analogs of Demazure characters) by constructing a K-theoretic analog of crystals with an appropriate analog of a Demazure crystal. We relate our crystal structure to combinatorial models using excited Young diagrams, Gelfand-Tsetlin patterns via the $5$-vertex model, and biwords via Hecke insertion to compute symmetric Grothendieck polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-09T17:58:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "05E10",
      "17B37",
      "14M15"
    ],
    "keywords": [
      "crystal structure",
      "grothendieck polynomials representing schubert classes",
      "symmetric grothendieck polynomials representing schubert",
      "combinatorial",
      "decomposing symmetric grothendieck polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}