{
  "id": "math/0703361",
  "version": "v2",
  "published": "2007-03-12T22:21:43.000Z",
  "updated": "2007-05-25T17:41:17.000Z",
  "title": "Coxeter Elements and Periodic Auslander-Reiten Quiver",
  "authors": [
    "Alexander Kirillov Jr.",
    "Jaimal Thind"
  ],
  "comment": "27 pages, 10 figures. v2: Added new sections relating our results to the theory of quiver representations",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "In this paper we show that for a simply-laced root system a choice of $C$ gives rise to a natural construction of the Dynkin diagram, in which vertices of the diagram correspond to $C$-orbits in $R$; moreover, it gives an identification of $R$ with a certain subset $Ihat$ of $I x Z_{2h}$, where $h$ is the Coxeter number. The set $Ihat$ has a natural quiver structure; we call it the periodic Auslander-Reiten quiver. This gives a combinatorial construction of the root system associated with the Dynkin diagram $I$: roots are vertices of $Ihat$, and the root lattice and the inner product admit an explicit description in terms of $Ihat$. Finally, we relate this construction to the theory of quiver representations.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-05-25T17:41:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "periodic auslander-reiten quiver",
      "coxeter elements",
      "dynkin diagram",
      "natural quiver structure",
      "inner product admit"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......3361K"
    }
  }
}