{
  "id": "1603.02521",
  "version": "v1",
  "published": "2016-03-08T14:01:03.000Z",
  "updated": "2016-03-08T14:01:03.000Z",
  "title": "The Upper Cluster Algebras of iARt Quivers I. Dynkin",
  "authors": [
    "Jiarui Fei"
  ],
  "comment": "48 pages. Comments are welcome. arXiv admin note: text overlap with arXiv:1210.1888 by other authors",
  "categories": [
    "math.RT",
    "math.AC",
    "math.CO",
    "math.RA"
  ],
  "abstract": "For each valued quiver $Q$ of Dynkin type, we construct a valued ice quiver $\\Delta_Q^2$ called the iARt quiver of $C^2Q$. Let $G$ be a simple connected Lie group with Dynkin diagram the underlying valued graph of $Q$. The upper cluster algebra of $\\Delta_Q^2$ is graded by the triple dominant weights $(\\mu,\\nu,\\lambda)$ of $G$. We prove that when $G$ is simply-laced, the dimension of each graded component counts the tensor multiplicity $c_{\\mu,\\nu}^\\lambda$. We conjecture that this is also true if $G$ is not simply-laced, and sketch a possible approach. Using this construction, we improve Berenstein-Zelevinsky's model, or in some sense generalize Knutson-Tao's hive model in type $A$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-08T14:01:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13F60",
      "16G20",
      "13A50",
      "52B20"
    ],
    "keywords": [
      "upper cluster algebra",
      "iart quiver",
      "sense generalize knutson-taos hive model",
      "simple connected lie group",
      "triple dominant weights"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160302521F"
    }
  }
}