{
  "id": "1009.3328",
  "version": "v2",
  "published": "2010-09-17T03:53:09.000Z",
  "updated": "2010-11-11T17:07:57.000Z",
  "title": "Geometric characterizations of the representation type of hereditary algebras and of canonical algebras",
  "authors": [
    "Calin Chindris"
  ],
  "comment": "27 pages. Fixed typos, changes/corrections to Section 6, few paragraphs about tame concealed algebras added",
  "categories": [
    "math.RT"
  ],
  "abstract": "We show that a finite connected quiver Q with no oriented cycles is tame if and only if for each dimension vector $\\mathbf{d}$ and each integral weight $\\theta$ of Q, the moduli space $\\mathcal{M}(Q,\\mathbf{d})^{ss}_{\\theta}$ of $\\theta$-semi-stable $\\mathbf{d}$-dimensional representations of Q is just a projective space. In order to prove this, we show that the tame quivers are precisely those whose weight spaces of semi-invariants satisfy a certain log-concavity property. Furthermore, we characterize the tame quivers as being those quivers Q with the property that for each Schur root $\\mathbf{d}$ of Q, the field of rational invariants $k(rep(Q,\\mathbf{d}))^{GL(\\mathbf{d})}$ is isomorphic to $k$ or $k(t)$. Next, we extend this latter description to canonical algebras. More precisely, we show that a canonical algebra $\\Lambda$ is tame if and only if for each generic root $\\mathbf{d}$ of $\\Lambda$ and each indecomposable irreducible component C of $rep(\\Lambda,\\mathbf{d})$, the field of rational invariants $k(C)^{GL(\\mathbf{d})}$ is isomorphic to $k$ or $k(t)$. Along the way, we establish a general reduction technique for studying fields of rational invariants on Schur irreducible components of representation varieties.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-11-11T17:07:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G20",
      "16G30",
      "16G60",
      "16G10"
    ],
    "keywords": [
      "canonical algebra",
      "representation type",
      "hereditary algebras",
      "geometric characterizations",
      "rational invariants"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.3328C"
    }
  }
}