{
  "id": "1602.05037",
  "version": "v1",
  "published": "2016-02-16T14:55:04.000Z",
  "updated": "2016-02-16T14:55:04.000Z",
  "title": "Classifying $τ$-tilting modules over the Auslander algebra of $K[x]/(x^n)$",
  "authors": [
    "Osamu Iyama",
    "Xiaojin Zhang"
  ],
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "We build a bijection between the set $\\sttilt\\Lambda$ of isomorphism classes of basic support $\\tau$-tilting modules over the Auslander algebra $\\Lambda$ of $K[x]/(x^n)$ and the symmetric group $\\mathfrak{S}_{n+1}$, which is an anti-isomorphism of partially ordered sets with respect to the generation order on $\\sttilt\\Lambda$ and the left order on $\\mathfrak{S}_{n+1}$. This restricts to the bijection between the set $\\tilt\\Lambda$ of isomorphism classes of basic tilting $\\Lambda$-modules and the symmetric group $\\mathfrak{S}_n$ due to Br\\\"{u}stle, Hille, Ringel and R\\\"{o}hrle. Regarding the preprojective algebra $\\Gamma$ of Dynkin type $A_n$ as a factor algebra of $\\Lambda$, we show that the tensor functor $-\\otimes_{\\Lambda}\\Gamma$ induces a bijection between $\\sttilt\\Lambda\\to\\sttilt\\Gamma$. This recover Mizuno's bijection $\\mathfrak{S}_{n+1}\\to\\sttilt\\Gamma$ for type $A_n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-16T14:55:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G10",
      "16E10"
    ],
    "keywords": [
      "auslander algebra",
      "tilting modules",
      "symmetric group",
      "isomorphism classes",
      "classifying"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160205037I"
    }
  }
}