{
  "id": "1410.3166",
  "version": "v1",
  "published": "2014-10-12T23:46:21.000Z",
  "updated": "2014-10-12T23:46:21.000Z",
  "title": "Irreducible components of varieties of representations I. The local case",
  "authors": [
    "Birge Huisgen-Zimmermann"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\Lambda$ be a local truncated path algebra over an algebraically closed field $K$, i.e., $\\Lambda$ is a quotient of a path algebra $KQ$ by the paths of length $L+1$, where $Q$ is the quiver with a single vertex and a finite number of loops and $L$ is a positive integer. For any $d>0$, we determine the irreducible components of the varieties that parametrize the $d$-dimensional representations of $\\Lambda$, namely, the components of the classical affine variety ${\\rm\\bf{Rep}}_{d}(\\Lambda)$ and -- equivalently -- those of the projective parametrizing variety ${\\rm GRASS}_d(\\Lambda)$. Our method is to corner the components by way of a twin pair of upper semicontinuous maps from ${\\rm\\bf{Rep}}_{d}(\\Lambda)$ to a poset consisting of sequences of semisimple modules. An excerpt of the main result is as follows. Given a sequence ${\\bf S} = ({\\bf S}_0, ..., {\\bf S}_L)$ of semisimple modules with $\\dim \\bigoplus_{0 \\le l \\le L} {\\bf S}_l = d$, let ${\\rm\\bf{Rep}}\\, {\\bf S}$ be the subvariety of ${\\rm\\bf{Rep}}_{d}(\\Lambda)$ consisting of the points that parametrize the modules with radical layering ${\\bf S}$. (The radical layering of a $\\Lambda$-module $M$ is the sequence $\\bigl(J^l M / J^{l+1} M\\bigr)_{0 \\le l \\le L}$, where $J$ is the Jacobson radical of $\\Lambda$.) Suppose the quiver $Q$ has $r \\ge 2$ loops. If $d \\le L+1$, the variety ${\\rm\\bf{Rep}}_{d}(\\Lambda)$ is irreducible. If, on the other hand, $d > L+1$, then the irreducible components of ${\\rm\\bf{Rep}}_{d}(\\Lambda)$ are the closures of the subvarieties ${\\rm\\bf{Rep}}\\, {\\bf S}$ for those sequences ${\\bf S}$ which satisfy the inequalities $\\dim {\\bf S}_l \\le r \\dim {\\bf S}_{l+1}$ and $\\dim {\\bf S}_{l+1} \\le r \\dim {\\bf S}_l$ for $0 \\le l < L$. As a byproduct, the main result provides generic information on the modules corresponding to the irreducible components of the parametrizing varieties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-12T23:46:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G10",
      "16G20",
      "14M15",
      "14M20"
    ],
    "keywords": [
      "irreducible components",
      "local case",
      "semisimple modules",
      "main result",
      "local truncated path algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.3166H"
    }
  }
}