{
  "id": "1503.08916",
  "version": "v1",
  "published": "2015-03-31T05:09:56.000Z",
  "updated": "2015-03-31T05:09:56.000Z",
  "title": "Newton-Okounkov bodies for Bott-Samelson varieties and string polytopes for generalized Demazure modules",
  "authors": [
    "Naoki Fujita"
  ],
  "comment": "24 pages",
  "categories": [
    "math.RT",
    "math.AG"
  ],
  "abstract": "Let $Z_{\\bf i}$ (resp., $X(w)$) be the Bott-Samelson variety (resp., the Schubert variety), and $\\mathcal{L}_{\\bf m}$ (resp., $\\mathcal{L}_\\lambda$) a line bundle on $Z_{\\bf i}$ (resp., on $X(w)$). We can think of $H^0(Z_{\\bf i}, \\mathcal{L}_{\\bf m})$ as a generalization of $H^0(X(w), \\mathcal{L}_\\lambda)$. We extend the string polytope for the Demazure module $H^0(X(w), \\mathcal{L}_\\lambda)^\\ast$ to $H^0(Z_{\\bf i}, \\mathcal{L}_{\\bf m})^\\ast$, and prove that it is identical to the Newton-Okounkov body of $Z_{\\bf i}$ with respect to a specific valuation. As applications of this result, we show that these are indeed polytopes, and also construct a basis of $H^0(Z_{\\bf i}, \\mathcal{L}_{\\bf m})$, which can be thought of as a perfect basis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-31T05:09:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "14M15",
      "14M25",
      "17B37"
    ],
    "keywords": [
      "generalized demazure modules",
      "bott-samelson variety",
      "newton-okounkov body",
      "string polytope",
      "schubert variety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150308916F"
    }
  }
}