{
  "id": "1101.1687",
  "version": "v3",
  "published": "2011-01-09T23:50:53.000Z",
  "updated": "2014-11-09T16:05:36.000Z",
  "title": "Crystal bases and Newton-Okounkov bodies",
  "authors": [
    "Kiumars Kaveh"
  ],
  "comment": "Numerous revisions were made. To appear in Duke Mathematical Journal. 33 pages",
  "categories": [
    "math.AG",
    "math.RT"
  ],
  "abstract": "Let G be a connected reductive algebraic group. We prove that the string parametrization of a crystal basis for a finite dimensional irreducible representation of G extends to a natural valuation on the field of rational functions on the flag variety G/B, which is a highest term valuation corresponding to a coordinate system on a Bott-Samelson variety. This shows that the string polytopes associated to irreducible representations, can be realized as Newton-Okounkov bodies for the flag variety. This is closely related to an earlier result of A. Okounkov for the Gelfand-Cetlin polytopes of the symplectic group. As a corollary we recover a multiplicativity property of the canonical basis due to P. Caldero. We generalize the results to spherical varieties. From these the existence of SAGBI bases for the homogeneous coordinate rings of flag and spherical varieties, as well as their toric degenerations follow, recovering previous results of Caldero, Alexeev-Brion and the author.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-09-26T15:35:07.000Z",
      "abstract": "Let G be a connected reductive algebraic group. We prove that the string parametrization of a crystal basis for a finite dimensional irreducible representation of G coincides with a natural valuation on the field of rational functions on the flag variety G/B, which is a highest term valuation corresponding to a coordinate system on a Bott-Samelson variety. This shows that the string polytopes associated to irreducible representations, can be realized as Newton-Okounkov bodies for the flag variety. This is closely related to an earlier result of A. Okounkov for the Gelfand-Cetlin polytopes of the symplectic group. As a corollary we deduce a multiplicativity property of the canonical basis due to P. Caldero. We generalize the results to spherical varieties. From these the existence of SAGBI bases for the homogeneous coordinate rings of flag and spherical varieties, as well as their toric degenerations follow recovering previous results of Caldero, Alexeev-Brion and the author.",
      "comment": "Totally revised. A statement in the previous version (Theorem 2.5 about highest term vs. lowest term valuations on Bott-Samelson variety) turned out to be false and has been removed in this version. A new section with an example of the main result added. 30 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-11-09T16:05:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M15",
      "05E10",
      "14M27"
    ],
    "keywords": [
      "newton-okounkov bodies",
      "crystal bases",
      "spherical varieties",
      "flag variety g/b",
      "finite dimensional irreducible representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1101.1687K"
    }
  }
}