{
  "id": "1711.00163",
  "version": "v1",
  "published": "2017-11-01T02:02:22.000Z",
  "updated": "2017-11-01T02:02:22.000Z",
  "title": "Cluster Algebras, Invariant Theory, and Kronecker Coefficients II",
  "authors": [
    "Jiarui Fei"
  ],
  "comment": "40 pages, 20 figures, comments are welcome",
  "categories": [
    "math.RT",
    "math.AC",
    "math.CO",
    "math.GR"
  ],
  "abstract": "We prove that the semi-invariant ring of the standard representation space of the $l$-flagged $m$-arrow Kronecker quiver is an upper cluster algebra for any $l,m\\in \\mathbb{N}$. The quiver and cluster are explicitly given. We prove that the quiver with its rigid potential is a polyhedral cluster model. As a consequence, to compute each Kronecker coefficient $g_{\\mu,\\nu}^\\lambda$ with $\\lambda$ at most $m$ parts, we only need to count lattice points in at most $m!$ fibre (rational) polytopes inside the ${\\rm g}$-vector cone, which is explicitly given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-01T02:02:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13F60",
      "20C30",
      "16G20",
      "13A50",
      "52B20"
    ],
    "keywords": [
      "kronecker coefficient",
      "invariant theory",
      "standard representation space",
      "arrow kronecker quiver",
      "upper cluster algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}