{
  "id": "1605.06330",
  "version": "v1",
  "published": "2016-05-20T12:47:42.000Z",
  "updated": "2016-05-20T12:47:42.000Z",
  "title": "Varieties of $G_r$-summands in Rational $G$-modules",
  "authors": [
    "Paul Sobaje"
  ],
  "comment": "17 pages, comments very welcome",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G$ be a simple simply connected algebraic group over an algebraically closed field $k$ of characteristic $p$, with $r$-th Frobenius kernel $G_r$. Let $M$ be a $G_r$-module and $V$ a rational $G$-module. We put a variety structure on the set of all $G_r$-summands of $V$ that are isomorphic to $M$, and study basic properties of these varieties. We give a few applications of this work to the representation theory of $G$, primarily in providing some sufficient conditions for when a $G_r$-module decomposition of $V$ can be extended to a $G$-module decomposition. In particular we are interested in connections to Donkin's tilting module conjecture, and more generally to the problem of finding a $G$-structure for the projective indecomposable $G_r$-modules. To that end, we show that Donkin's conjecture is equivalent to determining the linearizability or non-linearizability of $G$-actions on certain affine spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-20T12:47:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10"
    ],
    "keywords": [
      "module decomposition",
      "study basic properties",
      "simple simply connected algebraic group",
      "donkins tilting module conjecture",
      "th frobenius kernel"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}