{
  "id": "1504.02970",
  "version": "v1",
  "published": "2015-04-12T13:31:57.000Z",
  "updated": "2015-04-12T13:31:57.000Z",
  "title": "Cluster Algebras, Invariant Theory, and Kronecker Coefficients I",
  "authors": [
    "Jiarui Fei"
  ],
  "comment": "40 pages, 4 figures, comments are welcome",
  "categories": [
    "math.RT",
    "math.AC",
    "math.CO",
    "math.RA"
  ],
  "abstract": "We relate the $m$-truncated Kronecker products of symmetric functions to the semi-invariant rings of a family of quiver representations. We find cluster algebra structures for these semi-invariant rings when $m=2$. Each {\\sf g}-vector cone ${\\sf G}_{\\Diamond_l}$ of these cluster algebras controls the $2$-truncated Kronecker products for all symmetric functions of degree no greater than $l$. As a consequence, each relevant Kronecker coefficient is the difference of the number of the lattice points inside two rational polytopes. We also give explicit description of all ${\\sf G}_{\\Diamond_l}$'s. As an application, we compute some invariant rings.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-12T13:31:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C30",
      "13F60",
      "16G20",
      "13A50",
      "52B20"
    ],
    "keywords": [
      "invariant theory",
      "truncated kronecker products",
      "symmetric functions",
      "semi-invariant rings",
      "lattice points inside"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150402970F"
    }
  }
}