{
  "id": "1702.04206",
  "version": "v1",
  "published": "2017-02-14T13:41:31.000Z",
  "updated": "2017-02-14T13:41:31.000Z",
  "title": "Representations of regular trees and invariants of AR-components for generalized Kronecker quivers",
  "authors": [
    "Daniel Bissinger"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "We investigate the generalized Kronecker algebra $\\mathcal{K}_r = k\\Gamma_r$ with $r \\geq 3$ arrows. Given a regular component $\\mathcal{C}$ of the Auslander-Reiten quiver of $\\mathcal{K}_r$, we show that the quasi-rank $rk(\\mathcal{C}) \\in \\mathbb{Z}_{\\leq 1}$ can be described almost exactly as the distance $\\mathcal{W}(\\mathcal{C}) \\in \\mathbb{N}_0$ between two non-intersecting cones in $\\mathcal{C}$, given by modules with the equal images and the equal kernels property; more precisley, we show that the two numbers are linked by the inequality \\[ -\\mathcal{W}(\\mathcal{C}) \\leq rk(\\mathcal{C}) \\leq - \\mathcal{W}(\\mathcal{C}) + 3.\\] Utilizing covering theory, we construct for each $n \\in \\mathbb{N}_0$ a bijection $\\varphi_n$ between the field $k$ and $\\{ \\mathcal{C} \\mid \\mathcal{C} \\ \\text{regular component}, \\ \\mathcal{W}(\\mathcal{C}) = n \\}$. As a consequence, we get new results about the number of regular components of a fixed quasi-rank.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-14T13:41:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "generalized kronecker quivers",
      "regular trees",
      "regular component",
      "ar-components",
      "representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}