{
  "id": "1707.07740",
  "version": "v1",
  "published": "2017-07-24T20:28:41.000Z",
  "updated": "2017-07-24T20:28:41.000Z",
  "title": "On the Humphreys conjecture on support varieties of tilting modules",
  "authors": [
    "Pramod N. Achar",
    "William Hardesty",
    "Simon Riche"
  ],
  "comment": "54 pages, 2 color figures",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G$ be a simply-connected semisimple algebraic group over an algebraically closed field of characteristic $p$, assumed to be larger than the Coxeter number. The \"support variety\" of a $G$-module $M$ is a certain closed subvariety of the nilpotent cone of $G$, defined in terms of cohomology for the first Frobenius kernel $G_1$. In the 1990s, Humphreys proposed a conjectural description of the support varieties of tilting modules; this conjecture has been proved for $G = \\mathrm{SL}_n$ in earlier work of the second author. In this paper, we show that for any $G$, the support variety of a tilting module always contains the variety predicted by Humphreys, and that they coincide (i.e., the Humphreys conjecture is true) when $p$ is sufficiently large. We also prove variants of these statements involving \"relative support varieties.\"",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-24T20:28:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "support variety",
      "tilting module",
      "humphreys conjecture",
      "first frobenius kernel",
      "simply-connected semisimple algebraic group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 54,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}