{
  "id": "1907.11415",
  "version": "v1",
  "published": "2019-07-26T07:56:06.000Z",
  "updated": "2019-07-26T07:56:06.000Z",
  "title": "Crystal structures for canonical Grothendieck functions",
  "authors": [
    "Graham Hawkes",
    "Travis Scrimshaw"
  ],
  "comment": "23 pages, 1 figure",
  "categories": [
    "math.CO",
    "math.KT",
    "math.QA",
    "math.RT"
  ],
  "abstract": "We give a $U_q(\\mathfrak{sl}_n)$-crystal structure on multiset-valued tableaux, hook-valued tableaux, and valued-set tableaux, whose generating functions are the weak symmetric, canonical, and dual weak symmetric Grothendieck functions, respectively. We show the result is isomorphic to a (possibly infinite) direct sum of highest weight crystals, and for multiset-valued tableaux and valued-set tableaux, we provide an explicit bijection. As a consequence, these generating functions are Schur positive; in particular, the canonical Grothendieck functions, which was not previously known. We also give an extension of Hecke insertion to express a dual stable Grothenieck function as a sum of Schur functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-26T07:56:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "05A19",
      "14M15",
      "17B37"
    ],
    "keywords": [
      "canonical grothendieck functions",
      "crystal structure",
      "dual weak symmetric grothendieck functions",
      "valued-set tableaux",
      "highest weight crystals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}