Newton-Okounkov polytopes of Schubert varieties arising from cluster structures

Fujita, Naoki; Oya, Hironori

doi:10.48550/arxiv.2002.09912

Cited by 7 publications

(16 citation statements)

References 66 publications

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“…In this section, we review a construction of higher rank valuations using cluster structures, following [21]. Let us start with recalling the definition of upper cluster algebras of geometric type, following [3,16] but using the notation in [14,26].…”

Section: Cluster Algebras and Higher Rank Valuationsmentioning

confidence: 99%

“…Gross-Hacking-Keel-Kontsevich [26] developed the theory of cluster ensembles using methods in mirror symmetry, and showed that the theory of cluster algebras can be used to construct toric degenerations of projective varieties. The author and Oya [21] gave another approach to toric degenerations by relating the theory of cluster algebras with Newton-Okounkov bodies, and constructed a family of toric degenerations of flag varieties using cluster structures. Our aim of the present paper is to prove that the toric degenerations in this family induce semitoric degenerations of Richardson varieties.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 4, we recall some basic definitions on cluster algebras. We also see a construction of valuations using cluster structures, following [21]. In Section 5, we review cluster structures on unipotent cells, and recall results in [21].…”

Section: Introductionmentioning

confidence: 99%

“…We also see a construction of valuations using cluster structures, following [21]. In Section 5, we review cluster structures on unipotent cells, and recall results in [21]. In Section 6, we prove Theorem 1 above.…”

Section: Introductionmentioning

confidence: 99%

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Semi-toric degenerations of Richardson varieties arising from cluster structures on flag varieties

Fujita¹

2021

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A toric degeneration of an irreducible variety is a flat degeneration to an irreducible toric variety. In the case of a flag variety, its toric degeneration with desirable properties induces degenerations of Richardson varieties to unions of irreducible closed toric subvarieties, called semi-toric degenerations. For instance, Morier-Genoud proved that Caldero's toric degenerations arising from string polytopes have this property. Semi-toric degenerations are closely related to Schubert calculus. Indeed, Kogan-Miller constructed semi-toric degenerations of Schubert varieties from Knutson-Miller's semi-toric degenerations of matrix Schubert varieties which give a geometric proof of the pipe dream formula of Schubert polynomials. In this paper, we focus on a toric degeneration of a flag variety arising from a cluster structure, and prove that it induces semi-toric degenerations of Richardson varieties. Our semi-toric degeneration can be regarded as a generalization of Morier-Genoud's and Kogan-Miller's semi-toric degenerations.

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Section: Cluster Algebras and Higher Rank Valuationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Semi-toric degenerations of Richardson varieties arising from cluster structures on flag varieties

Fujita¹

2021

Preprint

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“…(i) Gelfand-Tsetlin polytopes [33,34], (ii) Berenstein-Littelmann-Zelevinsky's string polytopes [24], (iii) Nakashima-Zelevinsky polytopes [16], (iv) Lusztig polytopes [6], (v) Feigin-Fourier-Littelmann-Vinberg (FFLV) polytopes [11,27,28], (vi) polytopes constructed from extended g-vectors in cluster theory [18],…”

Section: Introductionmentioning

confidence: 99%

Newton-Okounkov polytopes of flag varieties and marked chain-order polytopes

Fujita¹

2021

Preprint

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Marked chain-order polytopes are convex polytopes constructed from a marked poset, which give a discrete family relating a marked order polytope with a marked chain polytope. In this paper, we consider the Gelfand-Tsetlin poset of type A, and realize the associated marked chain-order polytopes as Newton-Okounkov bodies of the flag variety. Our realization connects previous realizations of Gelfand-Tsetlin polytopes and Feigin-Fourier-Littelmann-Vinberg polytopes as Newton-Okounkov bodies in a uniform way. As an application, we prove that the flag variety degenerates into the irreducible normal projective toric variety corresponding to a marked chain-order polytope. We also construct a specific basis of an irreducible highest weight representation which is naturally parametrized by the set of lattice points in a marked chain-order polytope.

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Newton–Okounkov polytopes of flag varieties and marked chain-order polytopes

Fujita

2023

Trans. Amer. Math. Soc. Ser. B

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Marked chain-order polytopes are convex polytopes constructed from a marked poset. They give a discrete family relating a marked order polytope with a marked chain polytope. In this paper, we consider the Gelfand–Tsetlin poset of type A A , and realize the associated marked chain-order polytopes as Newton–Okounkov bodies of the flag variety. Our realization connects previous realizations of Gelfand–Tsetlin polytopes and Feigin–Fourier–Littelmann–Vinberg polytopes as Newton–Okounkov bodies in a uniform way. As an application, we prove that the flag variety degenerates into the irreducible normal projective toric variety corresponding to a marked chain-order polytope. We also construct a specific basis of an irreducible highest weight representation. The basis is naturally parametrized by the set of lattice points in a marked chain-order polytope.

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Newton-Okounkov polytopes of Schubert varieties arising from cluster structures

Cited by 7 publications

References 66 publications

Semi-toric degenerations of Richardson varieties arising from cluster structures on flag varieties

Semi-toric degenerations of Richardson varieties arising from cluster structures on flag varieties

Newton-Okounkov polytopes of flag varieties and marked chain-order polytopes

Newton–Okounkov polytopes of flag varieties and marked chain-order polytopes

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