{
  "id": "0912.0657",
  "version": "v2",
  "published": "2009-12-03T13:58:59.000Z",
  "updated": "2010-04-27T05:27:36.000Z",
  "title": "The Gabriel-Roiter measures of the indecomposables in a regular component of the 3-Kronecker quiver",
  "authors": [
    "Bo Chen"
  ],
  "comment": "this paper is combined with a new revised preprint arXiv:1001.4954.",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $Q$ be the 3-Kronecker quiver, i.e., $Q$ has two vertices, labeled by 1 and 2, and three arrows from 2 to 1. Fix an algebraically closed field $k$. Let $\\mathcal{C}$ be a regular component of the Auslander-Reiten quiver containing an indecomposable module $X$ with dimension $(1,1)$ or $(2,1)$. Using the properties of the Fibonacci numbers, we show that the Gabriel-Roiter measures of the indecomposable modules in $\\mathcal{C}$ are uniquely determined by the dimension vectors. In other words, two indecomposable modules in $\\mathcal{C}$ are not isomorphic if and only if their Gabriel-Roiter measures are different.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-04-27T05:27:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G20",
      "16G70"
    ],
    "keywords": [
      "gabriel-roiter measures",
      "regular component",
      "indecomposable module",
      "dimension vectors",
      "fibonacci numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.0657C"
    }
  }
}