{
  "id": "0812.1432",
  "version": "v1",
  "published": "2008-12-08T08:17:58.000Z",
  "updated": "2008-12-08T08:17:58.000Z",
  "title": "Polynomial Representation of $E_7$ and Its Combinatorial and PDE Implications",
  "authors": [
    "Xiaoping Xu"
  ],
  "comment": "37pages",
  "categories": [
    "math.RT",
    "math.QA"
  ],
  "abstract": "In this paper, we use partial differential equations to find the decomposition of the polynomial algebra over the basic irreducible module of $E_7$ into a sum of irreducible submodules. Moreover, we obtain a combinatorial identity, saying that the dimensions of certain irreducible modules of $E_7$ are correlated by the binomial coefficients of fifty-five. Furthermore, we prove that two families of irreducible submodules with three integral parameters are solutions of the fundamental invariant differential operator corresponding to Cartan's unique quartic $E_7$ invariant.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-12-08T08:17:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B25"
    ],
    "keywords": [
      "polynomial representation",
      "pde implications",
      "combinatorial",
      "fundamental invariant differential operator corresponding",
      "partial differential equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1245160,
      "adsabs": "2008arXiv0812.1432X"
    }
  }
}