{
  "id": "1610.03692",
  "version": "v1",
  "published": "2016-10-12T12:45:08.000Z",
  "updated": "2016-10-12T12:45:08.000Z",
  "title": "A $q$-Robinson-Schensted-Knuth Algorithm and a $q$-polymer",
  "authors": [
    "Yuchen Pei"
  ],
  "comment": "28 pages. Comments are welcome",
  "categories": [
    "math.CO",
    "math.PR",
    "math.QA",
    "math.RT"
  ],
  "abstract": "In [Matveev-Petrov 2016](arXiv:1504.00666) a $q$-deformed Robinson-Schensted-Knuth algorithm ($q$RSK) was introduced. In this article we give reformulations of this algorithm in terms of Noumi-Yamada description, growth diagrams and local moves. We show that the algorithm is symmetric, namely the output tableaux pair are swapped in a sense of distribution when the input matrix is transposed. We also formulate a $q$-polymer model based on the $q$RSK, prove the corresponding Burke property, which we use to show a strong law of large numbers for the partition function given stationary boundary conditions and $q$-geometric weights. We use the $q$-local moves to define a generalisation of $q$RSK taking a Young diagram-shape of array as the input. We write down the joint distribution of partition functions in the space-like direction of the $q$-polymer in $q$-geometric environment, formulate a $q$-version of the multilayer polynuclear growth model ($q$PNG) and write down the joint distribution of the $q$-polymer partition functions at a fixed time.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-12T12:45:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "robinson-schensted-knuth algorithm",
      "multilayer polynuclear growth model",
      "local moves",
      "joint distribution",
      "polymer partition functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}