{
  "id": "math/0606775",
  "version": "v1",
  "published": "2006-06-30T02:58:38.000Z",
  "updated": "2006-06-30T02:58:38.000Z",
  "title": "Semicanonical basis generators of the cluster algebra of type $A_1^{(1)}$",
  "authors": [
    "Andrei Zelevinsky"
  ],
  "comment": "4 pages, no figures",
  "categories": [
    "math.RA",
    "math.CO"
  ],
  "abstract": "We study the cluster variables and \"imaginary\" elements of the semicanonical basis for the coefficient-free cluster algebra of affine type $A_1^{(1)}$. A closed formula for the Laurent expansions of these elements was obtained by P.Caldero and the author in math.RT/0604054. As a by-product, there was given a combinatorial interpretation of the Laurent polynomials in question, equivalent to the one obtained by G.Musiker and J.Propp in math.CO/0602408. The arguments in math.RT/0604054 used a geometric interpretation of the Laurent polynomials due to P.Caldero and F.Chapoton (math.RT/0410184). This note provides a quick, self-contained and completely elementary alternative proof of the same results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-06-30T02:58:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16S99"
    ],
    "keywords": [
      "semicanonical basis generators",
      "laurent polynomials",
      "coefficient-free cluster algebra",
      "combinatorial interpretation",
      "affine type"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......6775Z"
    }
  }
}