Cones, crystals, and patterns

Littelmann, Peter

doi:10.1007/bf01236431

Cited by 217 publications

(427 citation statements)

References 17 publications

Supporting

Mentioning

419

Contrasting

Unclassified

Order By: Relevance

“…There is a notion of Gelfand-Tsetlin polytopes in types B, C and D. Their defining inequalities can be found in [BZ88] and [Lit98,§6,§7]. From the definitions, it is obvious that in types B n and C n the fan Σ w std 0 is trivial and the polytope Q w std 0 (λ) is integral whenever the λ, α ∨ i are all even.…”

Section: String and Moment Polytopesmentioning

confidence: 99%

Toric degenerations of spherical varieties

Alexeev

Brion

2005

Sel. math., New ser.

164

View full text Add to dashboard Cite

Abstract. We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by Mirror Symmetry, we give conditions for the limit toric variety to be a Gorenstein Fano, and provide many examples. We also provide an explanation for the limits as boundary points of the moduli space of stable pairs whose existence is predicted by the Minimal Model Program.

show abstract

Section: String and Moment Polytopesmentioning

confidence: 99%

Toric degenerations of spherical varieties

Alexeev

Brion

2005

Sel. math., New ser.

164

View full text Add to dashboard Cite

show abstract

“…0.2. 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 in [10] and made precise by A. Berenstein and A. Zelevinsky in [1].…”

mentioning

confidence: 99%

Toric degenerations of Schubert varieties

Caldero

2002

Transformation Groups

View full text Add to dashboard Cite

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

show abstract

“…Indeed, as explained in [3,15], there exist bases for highest weight representations corresponding to any reduced expression for the long element w 0 of the Weyl group of GL(r + 1) -S r , the symmetric group on r letters -as a product of simple reflections σ i . These make use of the Kashiwara crystal graph and are commonly called string bases.…”

Section: Metaplectic Whittaker Functions and Patternsmentioning

confidence: 99%

Metaplectic Ice

Brubaker¹,

Bump²,

Chinta³

et al. 2012

Multiple Dirichlet Series, L-Functions and Automorphic Forms

View full text Add to dashboard Cite

Abstract. We study spherical Whittaker functions on a metaplectic cover of GL(r + 1) over a nonarchimedean local field using lattice models from statistical mechanics. An explicit description of this Whittaker function was given in terms of Gelfand-Tsetlin patterns in [5,17], and we translate this description into an expression of the values of the Whittaker function as partition functions of a sixvertex model. Properties of the Whittaker function may then be expressed in terms of the commutativity of row transfer matrices potentially amenable to proof using the Yang-Baxter equation. We give two examples of this: first, the equivalence of two different Gelfand-Tsetlin definitions, and second, the effect of the Weyl group action on the Langlands parameters. The second example is closely connected with another construction of the metaplectic Whittaker function by averaging over a Weyl group action [9,10].

show abstract

Cones, crystals, and patterns

Cited by 217 publications

References 17 publications

Toric degenerations of spherical varieties

Toric degenerations of spherical varieties

Toric degenerations of Schubert varieties

Metaplectic Ice

Contact Info

Product

Resources

About