Toric degeneration of Schubert varieties and Gelfand¿Tsetlin polytopes

By Mikhail Kogan, Ezra Miller. 2003

120 citations recorded by the source corpus.

This note constructs the flat toric degeneration of the manifold Fln of flags in Cn due to Gonciulea and Lakshmibai (Transform. Groups 1(3) (1996) 215) as an explicit GIT quotient of the Grobner degeneration due to Knutson and Miller (Grobner geometry of Schubert polynomials, Ann. Math. (2) to appear). This implies that Schubert varieties degenerate to reduced unions of toric varieties, associated to faces indexed by rc-graphs (reduced pipe dreams) in the Gelfand–Tsetlin polytope. Our explicit description of the toric degeneration of Fln provides a simple explanation of how Gelfand–Tsetlin decompositions for irreducible polynomial representations of GLn arise via geometric quantization.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9bae2430f5f660d5460a1eba83e289244a1c621a

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Citations in the corpus

Combinatorial Commutative Algebra (2004, methodology): “28 are not coincidentally similar: there is a direct geometric connection [KoM04], in which each reduced pipe dream subspace from the Gröbner degeneration maps to a face of the Gelfand–Tsetlin toric variety.”
Chains in the Bruhat order (2005, methodology): “Remark 15.4. Toric degenerations of Schubert varieties Xw for Kempf elements (312-avoiding permutations in our terminology), were constructed by Gonciulea and Lakshmibai [GL], and were studied by Kogan [Kog] and Kogan-Miller [ KM ].”
Gröbner geometry of Schubert polynomials (2001, background): “determine cohomology classes whose sum is the Schubert class; nonetheless, their union is a cycle, and its class is the Schubert class. See also [ KoM03 ].”
Integrable systems, toric degenerations and Okounkov bodies (2012, methodology): “Several years ago, we had the idea to use the toric degeneration results for flag varieties given in Gonciulea– Lakshmibai [15], Kogan–Miller [30], and Caldero [9], in order to construct integrable systems for flag varieties of general reductive groups, thus generalizing the Gel’fand–Cetlin integrab”
Okounkov bodies and toric degenerations (2010, methodology): “6.1]. Indeed, GL3/B is isomorphic to P(TP2), the projectived tangent bundle of P2. Up to a lattice isomorphism, the polytope may also be identiﬁed with theGelfand-Tsetlin polytope, and itwas shown in [KM] thatthe ﬂagvarieties for GLn degenerate to these toric varieties. Finally, let M = ϕ∗L⊗pr∗ 1O(1”
Canonical bases for cluster algebras (2014, background): “olarized toric variety for Ξ ⊂Rn. See §9. We expect this specializes to a uniform construction of many degenerations of representation theoretic objects to toric varieties, see e.g., [C02], [AB], and [KM05]. Applied to the Fock-Goncharov moduli spaces of G-local systems, this will give for the ﬁrst ”
Toric degenerations of Gelfand-Cetlin systems and potential functions (2008, background): “We extend f : X → C to an (n−1)-parameter family to define the degeneration of flag manifolds in stages which is also introduced in [19], on which we will construct a toric degeneration of the Gelfand-Cetlin system.”