Toric degenerations of integrable systems on Grassmannians and polygon spaces

Nohara, Yuichi; Ueda, Kanji

doi:10.1215/00277630-2643839

Cited by 16 publications

(10 citation statements)

References 24 publications

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“…Inspired by ideas coming from the Gelfand-Cetlin system, D. Bouloc proved in 2015 [Bou15] a similar conjecture for all singular fibers of the Kapovich-Millson system of bending flows of 3D polygons [KM96], and also of another similar integrable system studied by Nohara and Ueda [NU14] on the 2-Grassmannian manifold. Namely, he showed that all these fibers are isotropic submanifolds (if the ambient symplectic variety itself is a manifold) or orbifolds (in special situations when the ambient symplectic spaces are orbifolds but not manifolds).…”

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confidence: 82%

Singular fibres of the Gelfand–Cetlin system on MANI( n )*

Bouloc

Miranda

Zung

2018

Phil. Trans. R. Soc. A.

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In this paper, we show that every singular fibre of the Gelfand-Cetlin system on co-adjoint orbits of unitary groups is a smooth isotropic submanifold which is diffeomorphic to a two-stage quotient of a compact Lie group by free actions of two other compact Lie groups. In many cases, these singular fibres can be shown to be homogeneous spaces or even diffeomorphic to compact Lie groups. We also give a combinatorial formula for computing the dimensions of all singular fibres, and give a detailed description of these singular fibres in many cases, including the so-called (multi-)diamond singularities. These (multi-)diamond singular fibres are degenerate for the Gelfand-Cetlin system, but they are Lagrangian submanifolds diffeomorphic to direct products of special unitary groups and tori. Our methods of study are based on different ideas involving complex ellipsoids, Lie groupoids and also general ideas coming from the theory of singularities of integrable Hamiltonian systems.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

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confidence: 82%

Singular fibres of the Gelfand–Cetlin system on MANI( n )*

Bouloc

Miranda

Zung

2018

Phil. Trans. R. Soc. A.

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“…For a Grassmannian Gr 2 (C n ), the plabic graphs are in bijection with triangulations of an n-gon, and in this case polytopes isomorphic to ours were obtained earlier by Nohara and Ueda. These polytopes were shown in [NU14] to be integral (unlike in the general case), and were used to construct toric degenerations of the Grassmannian Gr 2 (C n ), see also [BFF + 16].…”

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confidence: 99%

Newton–Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians

Rietsch¹,

Williams²

2019

Duke Math. J.

114

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In this article we use cluster structures and mirror symmetry to explicitly describe a natural class of Newton-Okounkov bodies for Grassmannians. We consider the Grassmannian X = Gr n−k (C n ), as well as the mirror dual Landau-Ginzburg model (X ○ , W ∶X ○ → C), whereX ○ is the complement of a particular anti-canonical divisor in a Langlands dual GrassmannianX = Gr k ((C n ) * ), and the superpotential W has a simple expression in terms of Plücker coordinates [MR13]. Grassmannians simultaneously have the structure of an A-cluster variety and an X -cluster variety [Sco06, Pos]; roughly speaking, a cluster variety is obtained by gluing together a collection of tori along birational maps [FZ02, FG06]. Given a plabic graph or, more generally, a cluster seed G, we consider two associated coordinate systems: a network orare the open positroid varieties in X andX, respectively. To each X -cluster chart Φ G and ample 'boundary divisor' D in X ∖ X ○ , we associate a Newton-Okounkov body ∆ G (D) in R k(n−k) , which is defined as the convex hull of rational points; these points are obtained from the multi-degrees of leading terms of the Laurent polynomials Φ * G (f ) for f on X with poles bounded by some multiple of D. On the other hand using the A-cluster chart Φ ∨ G on the mirror side, we obtain a set of rational polytopes -described in terms of inequalities -by writing the superpotential W as a Laurent polynomial in the A-cluster coordinates, and then "tropicalising". Our first main result is that the Newton-Okounkov bodies ∆ G (D) and the polytopes obtained by tropicalisation on the mirror side coincide. As an application, we construct degenerations of the Grassmannian to normal toric varieties corresponding to (dilates of) these Newton-Okounkov bodies. Our second main result is an explicit combinatorial formula in terms of Young diagrams, for the lattice points of the Newton-Okounkov bodies, in the case that the cluster seed G corresponds to a plabic graph. This formula has an interpretation in terms of the quantum Schubert calculus of Grassmannians [FW04].

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“…, r n ), which happens to be a manifold when r is generic. These moduli spaces of polygons and their bending systems have been studied from various points of views afterwards [5,6,11,12,13,17]. Our results here concern their singular fibers and state that the systems of bending flows on M r are indeed examples of systems with spherical singularities:…”

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confidence: 75%

“…After that, we prove in §5 that the singular fibers are isotropic. Finally in §6, we describe how the systems of bending flows on M r relate to integrable systems on the Grassmannian Gr(2, n) defined by Nohara and Ueda [17] and to the Gel'fand-Cetlin system on U (n). In particular we provide some arguments suggesting that the techniques employed in this paper would also apply to the integrable systems on Gr(2, n).…”

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confidence: 99%

Singular fibers of the bending flows on the moduli space of 3D polygons

Bouloc

2018

Journal of Symplectic Geometry

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In this paper, we prove that in the system of bending flows on the moduli space of polygons with fixed side lengths introduced by Kapovich and Millson, the singular fibers are isotropic homogeneous submanifolds. The proof covers the case where the system is defined by any maximal family of disjoint diagonals. We also take in account the case where the fixed side lengths are not generic. In this case, the phase space is an orbispace, and our result holds in the sense that singular fibers are isotropic orbispaces. In a last part we provide leads in favor of a similar study of the integrable systems defined by Nohara and Ueda on the Grassmannian of 2-planes in C n . ContentsAlso, we do not limit ourselves to the case when the side lengths are generic. When those lengths are chosen in such a way that the configuration space fails to be a manifold, it is still possible to work in the category of orbispaces. Using the concepts of tangent space, vector fields and symplectic structure on an orbispace (see e.g. [7,19, 20]), we extend the definition of the considered Hamiltonian systems to the non-generic case.Proposition B. When r is not generic, the moduli space M r is a symplectic orbispace, and the systems of bending flows still make sense.

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Toric degenerations of integrable systems on Grassmannians and polygon spaces

Cited by 16 publications

References 24 publications

Singular fibres of the Gelfand–Cetlin system on MANI( n )*

Singular fibres of the Gelfand–Cetlin system on MANI( n )*

Newton–Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians

Singular fibers of the bending flows on the moduli space of 3D polygons

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