{
  "id": "0712.1381",
  "version": "v2",
  "published": "2007-12-10T00:56:14.000Z",
  "updated": "2009-02-14T04:38:21.000Z",
  "title": "Cluster combinatorics of d-cluster categories",
  "authors": [
    "Yu Zhou",
    "Bin Zhu"
  ],
  "comment": "correted many typos according to the referee's comments, final version to appear in J. Algebra",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "We study the cluster combinatorics of $d-$cluster tilting objects in $d-$cluster categories. By using mutations of maximal rigid objects in $d-$cluster categories which are defined similarly for $d-$cluster tilting objects, we prove the equivalences between $d-$cluster tilting objects, maximal rigid objects and complete rigid objects. Using the chain of $d+1$ triangles of $d-$cluster tilting objects in [IY], we prove that any almost complete $d-$cluster tilting object has exactly $d+1$ complements, compute the extension groups between these complements, and study the middle terms of these $d+1$ triangles. All results are the extensions of corresponding results on cluster tilting objects in cluster categories established in [BMRRT] to $d-$cluster categories. They are applied to the Fomin-Reading's generalized cluster complexes of finite root systems defined and studied in [FR2] [Th] [BaM1-2], and to that of infinite root systems [Zh3].",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-02-14T04:38:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G20",
      "16G70",
      "05A15"
    ],
    "keywords": [
      "cluster tilting object",
      "cluster combinatorics",
      "d-cluster categories",
      "maximal rigid objects",
      "complete rigid objects"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0712.1381Z"
    }
  }
}