{
  "id": "1309.7301",
  "version": "v2",
  "published": "2013-09-27T17:22:01.000Z",
  "updated": "2014-12-10T16:12:05.000Z",
  "title": "A categorification of Grassmannian cluster algebras",
  "authors": [
    "Bernt Tore Jensen",
    "Alastair King",
    "Xiuping Su"
  ],
  "comment": "Minor change in title; new Sec 9 on categorification; elsewhere some changes in exposition and some new figures",
  "categories": [
    "math.RT"
  ],
  "abstract": "We describe a ring whose category of Cohen-Macaulay modules provides an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of k-planes in n-space. More precisely, there is a cluster character defined on the category which maps the rigid indecomposable objects to the cluster variables and the maximal rigid objects to clusters. This is proved by showing that the quotient of this category by a single projective-injective object is Geiss-Leclerc-Schroer's category Sub $Q_k$, which categorifies the coordinate ring of the big cell in this Grassmannian.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-09-27T17:22:01.000Z",
      "title": "A category for Grassmannian cluster algebras",
      "abstract": "We describe a ring whose category of Cohen-Macaulay modules `categorifies' the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of k-planes in n-space, in the sense that the rigid indecomposable objects in this category correspond to cluster variables and the maximal rigid objects to clusters. This is proved by showing that the quotient of this category by a single projective-injective object is Geiss-Leclerc-Schroer's category Sub $Q_k$, which categorifies the coordinate ring of the big cell in this Grassmannian.",
      "comment": null,
      "journal": null,
      "doi": null,
      "authors": [
        "Bernt Jensen",
        "Alastair King",
        "Xiuping Su"
      ]
    },
    {
      "version": "v2",
      "updated": "2014-12-10T16:12:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13F60",
      "16G50"
    ],
    "keywords": [
      "grassmannian cluster algebras",
      "cluster algebra structure",
      "geiss-leclerc-schroers category sub",
      "maximal rigid objects",
      "cohen-macaulay modules"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.7301T"
    }
  }
}