{
  "id": "1702.06447",
  "version": "v1",
  "published": "2017-02-21T15:46:02.000Z",
  "updated": "2017-02-21T15:46:02.000Z",
  "title": "Fields of definition for representations of associative algebras",
  "authors": [
    "Dave Benson",
    "Zinovy Reichstein"
  ],
  "comment": "12 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "We examine situations, where representations of a finite-dimensional $F$-algebra $A$ defined over a separable extension field $K/F$, have a unique minimal field of definition. Here the base field $F$ is assumed to be a $C_1$-field. In particular, $F$ could be a finite field or $k(t)$ or $k((t))$,where $k$ is algebraically closed. We show that a unique minimal field of definition exists if (a) $K/F$ is an algebraic extension or (b) $A$ is of finite representation type. Moreover, in these situations the minimal field of definition is a finite extension of $F$. This is not the case if $A$ is of infinite representation type or $F$ fails to be $C_1$. As a consequence, we compute the essential dimension of the functor of representations of a finite group, generalizing a theorem of N. Karpenko, J. Pevtsova and the second author.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-21T15:46:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G10",
      "16G60",
      "20C05"
    ],
    "keywords": [
      "definition",
      "associative algebras",
      "unique minimal field",
      "infinite representation type",
      "second author"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}