{
  "id": "1811.05501",
  "version": "v1",
  "published": "2018-11-13T19:15:49.000Z",
  "updated": "2018-11-13T19:15:49.000Z",
  "title": "A combinatorial $\\mathfrak{sl}_2$-action and the Sperner property for the weak order",
  "authors": [
    "Christian Gaetz",
    "Yibo Gao"
  ],
  "comment": "6 pages",
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "We construct a simple combinatorially-defined representation of $\\mathfrak{sl}_2$ which respects the order structure of the weak order on the symmetric group. This is used to resolve a conjecture of Stanley that the weak order has the strong Sperner property, and is therefore a Peck poset.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-13T19:15:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weak order",
      "strong sperner property",
      "order structure",
      "simple combinatorially-defined representation",
      "symmetric group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}