{
  "id": "1812.06711",
  "version": "v1",
  "published": "2018-12-17T11:30:46.000Z",
  "updated": "2018-12-17T11:30:46.000Z",
  "title": "Steinberg squares and tensor products of tilting modules with simple modules",
  "authors": [
    "Paul Sobaje"
  ],
  "comment": "8 pages, feedback very welcome",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G$ be a simple and simply connected algebraic group over an algebraically closed field $\\Bbbk$ of characteristic $p>0$. We establish an isomorphism of $G$-modules between a direct sum of modules $\\text{St} \\otimes \\text{St}$ and a direct sum of tensor products of simple modules of restricted highest weight with tilting modules that are projective over the Frobenius kernel of $G$. This isomorphism holds precisely when Donkin's Tilting Module Conjecture does, and thus can be seen as providing a $G$-module theoretic characterization of this conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-17T11:30:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "simple modules",
      "tensor products",
      "steinberg squares",
      "direct sum",
      "module theoretic characterization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}