{
  "id": "0708.3949",
  "version": "v3",
  "published": "2007-08-29T12:43:40.000Z",
  "updated": "2008-03-06T07:45:48.000Z",
  "title": "Categorification of Wedderburn's basis for \\mathbb{C}[S_n]",
  "authors": [
    "Volodymyr Mazorchuk",
    "Catharina Stroppel"
  ],
  "comment": "11 pages, some corrections, to appear in Arch. Math",
  "journal": "Arch. Math. (Basel) 91 (2008), no. 1, 1--11.",
  "categories": [
    "math.RT",
    "math.GR"
  ],
  "abstract": "M. Neunh{\\\"o}ffer studies in \\cite{Ne} a certain basis of $\\mathbb{C}[S_n]$ with the origins in \\cite{Lu} and shows that this basis is in fact Wedderburn's basis. In particular, in this basis the right regular representation of $S_n$ decomposes into a direct sum of irreducible representations (i.e. Specht or cell modules). In the present paper we rediscover essentially the same basis with a categorical origin coming from projective-injective modules in certain subcategories of the BGG-category $\\mathcal{O}$. An important role in our arguments is played by the dominant projective module in each of these categories. As a biproduct of the study of this dominant projective module we show that {\\it Kostant's problem} (\\cite{Jo}) has a negative answer for some simple highest weight module over the Lie algebra $\\mathfrak{sl}_4$, which disproves the general belief that Kostant's problem should have a positive answer for all simple highest weight modules in type $A$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2008-03-06T07:45:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "20B30"
    ],
    "keywords": [
      "simple highest weight module",
      "dominant projective module",
      "kostants problem",
      "categorification",
      "fact wedderburns basis"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0708.3949M"
    }
  }
}