{
  "id": "2007.11721",
  "version": "v1",
  "published": "2020-07-23T00:13:53.000Z",
  "updated": "2020-07-23T00:13:53.000Z",
  "title": "Perforated Tableaux: A Combinatorial Model for Crystal Graphs in Type $A_{n-1}$",
  "authors": [
    "Glenn D. Appleby",
    "Tamsen Whitehead"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We present a combinatorial model, called \\emph{perforated tableaux}, to study $A_{n-1}$ crystals, unifying several previously studied combinatorial models. We identify nodes in the $k$-fold tensor product of the standard crystal with length $k$ words in $[n]= \\{ 1, \\ldots n\\}$. We model this crystal with perforated tableaux (ptableaux) with simpler crystal operators with which we can identify highest weights visually without computation (for all crystals directly, without reference to a canonical model of semistandard Young tableaux (SSYT)). We generalize the tensor products in the Littlewood-Richardson rule to all of $[n]^{\\otimes k}$, and not just the irreducible crystals whose reading words come from SSYT. We relate evacuation (Lusztig involution) to products of ptableaux crystal operators, and find a combinatorial algorithm to compute commutators of highest weight ptableaux.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-23T00:13:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10"
    ],
    "keywords": [
      "combinatorial model",
      "perforated tableaux",
      "crystal graphs",
      "ptableaux crystal operators",
      "simpler crystal operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}