{
  "id": "1811.02237",
  "version": "v1",
  "published": "2018-11-06T09:13:29.000Z",
  "updated": "2018-11-06T09:13:29.000Z",
  "title": "Laurent phenomenon and simple modules of quiver Hecke algebras",
  "authors": [
    "Masaki Kashiwara",
    "Myungho Kim"
  ],
  "comment": "38 pages",
  "categories": [
    "math.RT",
    "math.QA"
  ],
  "abstract": "We study consequences of a monoidal categorification of the unipotent quantum coordinate ring $A_q(\\mathfrak{n}(w))$ together with the Laurent phenomenon of cluster algebras. We show that if a simple module $S$ in the category $\\mathcal C_w$ strongly commutes with all the cluster variables in a cluster $[ \\mathscr C]$, then $[S]$ is a cluster monomial in $[ \\mathscr C ]$. If $S$ strongly commutes with cluster variables except exactly one cluster variable $[M_k]$, then $[S]$ is either a cluster monomial in $[\\mathscr C ]$ or a cluster monomial in $\\mu_k([ \\mathscr C ])$. We give a new proof of the fact that the upper global basis is a common triangular basis (in the sense of Fan Qin) of the localization $\\widetilde A_q(\\mathfrak{n}(w))$ of $A_q(\\mathfrak{n}(w))$ at the frozen variables. A characterization on the commutativity of a simple module $S$ with cluster variables in a cluster $[ \\mathscr C]$ is given in terms of the denominator vector of $[S]$ with respect to the cluster $[ \\mathscr C]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-06T09:13:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13F60",
      "81R50",
      "17B37"
    ],
    "keywords": [
      "simple module",
      "quiver hecke algebras",
      "laurent phenomenon",
      "cluster variable",
      "cluster monomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}