{
  "id": "0803.3764",
  "version": "v2",
  "published": "2008-03-26T16:23:19.000Z",
  "updated": "2009-01-28T13:48:20.000Z",
  "title": "Cohomology and generic cohomology of Specht modules for the symmetric group",
  "authors": [
    "David J. Hemmer"
  ],
  "comment": "Some substantial revisions from previous version",
  "categories": [
    "math.RT",
    "math.GR"
  ],
  "abstract": "Cohomology of Specht modules for the symmetric group can be equated in low degrees with corresponding cohomology for the Borel subgroup B of the general linear group GL_d(k), but this has never been exploited to prove new symmetric group results. Using work of Doty on the submodule structure of symmetric powers of the natural GL_d(k) module together with work of Andersen on cohomology for B and its Frobenius kernels, we prove new results about H^i(\\Sigma_d, S^\\lambda). We recover work of James in the case i=0. Then we prove two stability theorems, one of which is a \"generic cohomology\" result for Specht modules equating cohomology of S^{p\\lambda} with S^{p^2\\lambda}. This is the first theorem we know relating Specht modules S^\\lambda and S^{p\\lambda}. The second result equates cohomology of S^\\lambda with S^{\\lambda + p^a\\mu} for large a.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-01-28T13:48:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C30",
      "20G10"
    ],
    "keywords": [
      "generic cohomology",
      "second result equates cohomology",
      "general linear group",
      "specht modules equating cohomology",
      "symmetric group results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0803.3764H"
    }
  }
}