{
  "id": "1908.08432",
  "version": "v1",
  "published": "2019-08-22T15:08:35.000Z",
  "updated": "2019-08-22T15:08:35.000Z",
  "title": "On the cohomology of line bundles over certain flag schemes II",
  "authors": [
    "Linyuan Liu",
    "Patrick Polo"
  ],
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "Over a field $K$ of characteristic $p$, let $Z$ be the incidence variety in $\\mathbb{P}^d \\times (\\mathbb{P}^d)^*$ and let $\\mathcal{L}$ be the restriction to $Z$ of the line bundle $\\mathcal{O}(-n-d) \\boxtimes \\mathcal{O}(n)$, where $n = p+f$ with $0 \\leq f \\leq p-2$. We prove that $H^d(Z,\\mathcal{L})$ is the simple $\\operatorname{GL}_{d+1}$-module corresponding to the partition $\\lambda_0 = (p-1+f,p-1,f+1)$. When $f= 0$, using the first author's description of $H^d(Z,\\mathcal{L})$ and Jantzen's sum formula, we obtain as a by-product that the sum of the monomial symmetric functions $m_\\lambda$, for all partitions $\\lambda$ of $2p-1$ less than $(p-1,p-1,1)$ in the dominance order, is the alternating sum of the Schur functions $S_{p-1,p-1-i,1^{i+1}}$ for $i=0,\\dots,p-2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-22T15:08:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "05E10",
      "14L15",
      "20G05"
    ],
    "keywords": [
      "line bundle",
      "flag schemes",
      "cohomology",
      "monomial symmetric functions",
      "jantzens sum formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}