{
  "id": "1405.0318",
  "version": "v2",
  "published": "2014-05-01T22:31:32.000Z",
  "updated": "2014-05-07T19:47:02.000Z",
  "title": "Generic representation theory of finite fields in nondescribing characteristic",
  "authors": [
    "Nicholas J. Kuhn"
  ],
  "comment": "11 pages revised version: reference to Helmstutler's work is updated",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let Rep(F;K) denote the category of functors from finite dimensional F-vector spaces to K-modules, where F is a field and K is a commutative ring. We prove that, if F is a finite field, and Char F is invertible in K, then the K-linear abelian category Rep(F;K) is equivalent to the product, over all k=0,1,2, ..., of the categories of K[GL(k,F)]-modules. As a consequence, if K is also a field, then small projectives are also injective in Rep(F;K), and will have finite length. Even more is true if Char K = 0: the category Rep(F;K) will be semisimple. In a last section, we briefly discuss \"q=1\" analogues and consider representations of various categories of finite sets. The main result follows from a 1992 result by L.G.Kovacs about the semigroup ring K[M_n(\\F)].",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-05-07T19:47:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "18A25",
      "20G05",
      "20M25",
      "16D90"
    ],
    "keywords": [
      "generic representation theory",
      "finite field",
      "nondescribing characteristic",
      "finite dimensional f-vector spaces",
      "k-linear abelian category rep"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.0318K"
    }
  }
}