{
  "id": "math/0012165",
  "version": "v1",
  "published": "2000-12-18T16:32:17.000Z",
  "updated": "2000-12-18T16:32:17.000Z",
  "title": "Toric degenerations of Schubert varieties",
  "authors": [
    "Philippe Caldero"
  ],
  "comment": "12 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G$ be a simply connected semi-simple complex algebraic group. Fix a maximal torus $T$ and a Borel subgroup $B$ such that $T\\subset B\\subset G$. Let $W$ the Weyl group of $G$ relative to $T$. For any $w$ in $W$, let $X_w=\\bar {BwB/B}$ denote the Schubert variety corresponding to $w$. This talk is concerned with the following problem : Is there a flat family over Spec${\\bf C}[t]$, such that the general fiber is $X_w$ and the special fiber is a toric variety? Our approach of the problem is based on the canonical/global base of Lusztig/Kashiwara and the so-called string parametrization of this base studied by P. Littelmann and made precise by A. Berenstein and A. Zelevinsky. Fix $w$ in $W$ and let $P^+$ be the semigroup of dominant weights. For all $\\lambda$ in $P^+$, let ${\\cal L}_\\lambda$ be the line bundle on $G/B$ corresponding to $\\lambda$. Then, the direct sum of global sections $R_w:=\\bigoplus_{\\lambda\\in P^+}H^0(X_w,{\\cal L}_\\lambda)$ carries a natural structure of $P^+$-graded ${\\bf C}$-algebra. Moreover, there exists a natural action of $T$ on $R_w$. Our principal result can be stated as follows : There exists a filtration $({\\cal F}_m^w)_{m\\in{\\bf N}}$ of $R_w$ such that (i) for all $m$ in ${\\bf N}$, ${\\cal F}_m^w$ is compatible with the $P^+$-grading of $R_w$, (ii) for all $m$ in ${\\bf N}$, ${\\cal F}_m^w$ is compatible with the action of $T$, (iii) the associated graded algebra is the ${\\bf C}$-algebra of the semigroup of integral points in a rational convex polyhedral cone. Equations for this cone were obtained by A. Berenstein and A. Zelevinski from $\\tilde w_0$-trails in fundamental Weyl modules of the Langlands dual of $G$. By standard arguments, the previous theorem gives a positive answer to the Degeneration Problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2000-12-18T16:32:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "schubert variety",
      "toric degenerations",
      "connected semi-simple complex algebraic group",
      "rational convex polyhedral cone",
      "fundamental weyl modules"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math.....12165C"
    }
  }
}