{
  "id": "1407.2296",
  "version": "v1",
  "published": "2014-07-08T23:26:07.000Z",
  "updated": "2014-07-08T23:26:07.000Z",
  "title": "Varieties of uniserial representations IV. Kinship to geometric quotients",
  "authors": [
    "Klaus Bongartz",
    "Birge Huisgen-Zimmermann"
  ],
  "journal": "Trans. Amer. Math. Soc. 353 (2001) 2091-2113",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "Let $\\Lambda$ be a finite dimensional algebra over an algebraically closed field, and ${\\Bbb S}$ a finite sequence of simple left $\\Lambda$-modules. In [6, 9], quasiprojective algebraic varieties with accessible affine open covers were introduced, for use in classifying the uniserial representations of $\\Lambda$ having sequence ${\\Bbb S}$ of consecutive composition factors. Our principal objectives here are threefold: One is to prove these varieties to be `good approximations' -- in a sense to be made precise -- to geometric quotients of the classical varieties $\\operatorname{Mod-Uni}({\\Bbb S})$ parametrizing the pertinent uniserial representations, modulo the usual conjugation action of the general linear group. To some extent, this fills the information gap left open by the frequent non-existence of such quotients. A second goal is that of facilitating the transfer of information among the `host' varieties into which the considered uniserial varieties can be embedded. These tools are then applied towards the third objective, concerning the existence of geometric quotients: We prove that $\\operatorname{Mod-Uni}({\\Bbb S})$ has a geometric quotient by the $GL$-action precisely when the uniserial variety has a geometric quotient modulo a certain natural algebraic group action, in which case the two quotients coincide. Our main results are exploited in a representation-theoretic context: Among other consequences, they yield a geometric characterization of the algebras of finite uniserial type which supplements existing descriptions, but is cleaner and more readily checkable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-08T23:26:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G10",
      "16G20",
      "16G60",
      "16P10"
    ],
    "keywords": [
      "information gap left open",
      "natural algebraic group action",
      "uniserial variety",
      "finite uniserial type",
      "finite dimensional algebra"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Trans. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.2296B"
    }
  }
}