{
  "id": "0811.1399",
  "version": "v1",
  "published": "2008-11-10T05:33:44.000Z",
  "updated": "2008-11-10T05:33:44.000Z",
  "title": "Polynomial Representation of E6 and Its Combinatorial and PDE Implications",
  "authors": [
    "Xiaoping Xu"
  ],
  "comment": "24pages",
  "categories": [
    "math.RT",
    "math.AP"
  ],
  "abstract": "In this paper, we use partial differential equations to find the decomposition of the polynomial algebra over the basic irreducible module of E6 into a sum of irreducible submodules. It turns out that the cubic polynomial invariant corresponding to the Dicksons' invariant trilinear form is the unique fundamental invariant. Moreover, we obtain a combinatorial identity saying that the dimensions of certain irreducible modules of E6 are correlated by the binomial coefficients of twenty-six. Furthermore, we find all the polynomial solutions for the invariant differential operator corresponding to the Dickson trilinear form in terms of the irreducible submodules.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-11-10T05:33:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B25"
    ],
    "keywords": [
      "pde implications",
      "polynomial representation",
      "combinatorial",
      "dickson trilinear form",
      "invariant trilinear form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0811.1399X"
    }
  }
}