{
  "id": "1610.03535",
  "version": "v1",
  "published": "2016-10-11T21:06:55.000Z",
  "updated": "2016-10-11T21:06:55.000Z",
  "title": "Pattern avoidance and fiber bundle structures on Schubert varieties",
  "authors": [
    "Timothy Alland",
    "Edward Richmond"
  ],
  "comment": "16 pages and 9 Figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "We give a permutation pattern avoidance criteria for determining when the projection map from the flag variety to a Grassmannian induces a fiber bundle structure on a Schubert variety. In particular, we introduce the notion of a split pattern and show that a Schubert variety has such a fiber bundle structure if and only if the corresponding permutation avoids the split patterns 3|12 and 23|1. Continuing, we show that a Schubert variety is an iterated fiber bundle of Grassmannian Schubert varieties if and only if the corresponding permutation avoids (non-split) patterns 3412, 52341, and 635241. This extends a combined result of Lakshmibai-Sandhya, Ryan, and Wolper who prove that Schubert varieties whose permutation avoids the \"smooth\" patterns 3412 and 4231 are iterated fiber bundles of smooth Grassmannian Schubert varieties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-11T21:06:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M15",
      "22E40",
      "20F55",
      "05A05",
      "05E05"
    ],
    "keywords": [
      "schubert variety",
      "fiber bundle structure",
      "corresponding permutation avoids",
      "iterated fiber bundle",
      "smooth grassmannian schubert varieties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}