Toric degenerations of Schubert varieties

Abstract

ABSTRACT. Let G be a simply connected semi-simple complex algebraic group. We prove that every Schubert variety of G has a flat degeneration into a toric variety. This provides a generalization of results of [7], [6], [5]. Our basic tool is Lusztig's canonical basis and the string parametrization of this basis. This research has been partially supported by the EC TMR network "Algebraic Lie Representations" , contract no. ERB FMTX-CT97-0100. 0. Introduction. 0.1. Let G be a simply connected semi-simple complex algebraic group. Fix a maximal torus T and a Borel subgroup B such that T ⊂ B ⊂ G. Let W the Weyl group of G relative to T . For any w in W , let X w = BwB/B denote the Schubert variety corresponding to w. This article is concerned with the following problem. Degeneration Problem. Is there a flat family over SpecC[t], such that the general fiber is X w and the special fiber is a toric variety?The existence of such a degeneration was obtained by N. Gonciulea and V. Lakshmibai for G = SL n , [7]. Their proof is based on the theory of standard monomials. In the case G = SL n , the corner stone of their proof is the following : fundamental weights are minuscule weights, hence, a basis of every fundamental representation is endowed with a structure of distributive lattice.A toric degeneration for Schubert varieties is given in [5], [6], for G of rank 2. The proofs rely on the theory of standard monomials as well. A natural question would be : is there a (flat) toric degeneration of the flag variety G/B which restricts to a toric

Degree bounds for type-A weight rings and Gelfand–Tsetlin semigroups

“…These are all straightforward consequences of the closed-form expressions (5.2) and (5.3) for T (1) j and T (2) j , respectively, so x is a GT-pattern. Thus, to show that x ∈ GT(λ, μ), we need only establish that the row-sums of x are correct.…”

“…We had hoped to use the same method in the case of n points in the projective plane, but the second theorem indicates why this is not the right approach. However, there might still be another toric degeneration, perhaps among those discovered by Caldero [2], that yields a bound better than the one in Theorem 3.9…”

“…We will construct a triangular array x by filling in the entries of x in blocks from the upper left to the lower right using the values T (1) j and T (2) j . Begin by filling the entries in the upper left of the triangular array as follows:…”

“…The basic idea behind the proof is to show that the degree-one piece contains a system of parameters and that the a-invariant of R(λ, μ) is negative. 2 (The a-invariant is the degree of the Hilbert series, which is a rational function.) The theorem then follows from the fact that R(λ, μ) is Cohen-Macaulay.…”

“…Here one takes λ to be a multiple of the second fundamental weight 2 . It was shown in [12] that the associated semigroup of Gelfand-Tsetlin patterns is generated in degree ≤2.…”