{
  "id": "1710.08754",
  "version": "v1",
  "published": "2017-10-24T13:21:50.000Z",
  "updated": "2017-10-24T13:21:50.000Z",
  "title": "Modular representations in type A with a two-row nilpotent central character",
  "authors": [
    "Vinoth Nandakumar",
    "David Yang"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "We study the category of representations of $\\mathfrak{sl}_{m+2n}$ in positive characteristic, whose p-character is a nilpotent whose Jordan type is the two-row partition (m+n,n). In a previous paper with Anno, we used Bezrukavnikov-Mirkovic-Rumynin's theory of positive characteristic localization and exotic t-structures to give a geometric parametrization of the simples using annular crossingless matchings. Building on this, here we give combinatorial dimension formulae for the simple objects, and compute the Jordan-Holder multiplicities of the simples inside the baby Vermas (in special case where n=1, i.e. that a subregular nilpotent, these were known from work of Jantzen). We use Cautis-Kamnitzer's geometric categorification of the tangle calculus to study the images of the simple objects under the [BMR] equivalence. The dimension formulae may be viewed as a positive characteristic analogue of the combinatorial character formulae for simple objects in parabolic category O for $\\mathfrak{sl}_{m+2n}$, due to Lascoux and Schutzenberger.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-24T13:21:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E47",
      "14M15",
      "14L35"
    ],
    "keywords": [
      "two-row nilpotent central character",
      "modular representations",
      "simple objects",
      "positive characteristic",
      "combinatorial dimension formulae"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}