Convexity and Thimm’s Trick

Lane, Jeremy

doi:10.1007/s00031-017-9436-7

Cited by 9 publications

(5 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We notice that some related partial results on the topology of collective integrable systems have been obtained by Lane in [Lan17]. We remark also that, even though the singularities of the Gelfand-Cetlin system are rather special from the point of view of general integrable Hamiltonian systems, there are still many similarities with other singularities that we encountered before.…”

supporting

confidence: 56%

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

Bouloc

Miranda

Zung

2018

Phil. Trans. R. Soc. A.

View full text Add to dashboard Cite

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'.

show abstract

supporting

confidence: 56%

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

Bouloc

Miranda

Zung

2018

Phil. Trans. R. Soc. A.

View full text Add to dashboard Cite

show abstract

“…In the examples of Section 7 we explain how our construction produces the systems studied in [HK15] and [HMM11] and relates these systems to the tropical Grassmannian variety of Speyer and Sturmfels [SS04]. Several aspects of recent work of Lane [Lan15] on the Hamiltonian geometry of Thimm's trick are related to our results. In particular, one can view our contraction map Φ X : X → X sc as a continuous extension of Thimm's trick from the principal subspace X I ⊂ X to all of X.…”

Section: Introductionmentioning

confidence: 90%

Contraction of Hamiltonian $K$-Spaces

Hilgert

Manon

Martens

2016

Int Math Res Notices

View full text Add to dashboard Cite

In the spirit of recent work of Harada-Kaveh and Nishinou-Nohara-Ueda, we study the symplectic geometry of Popov's horospherical degenerations of complex algebraic varieties with the action of a complex linearly reductive group. We formulate an intrinsic symplectic contraction of a Hamiltonian space, which is a surjective, continuous map onto a new Hamiltonian space that is a symplectomorphism on an explicitly defined dense open subspace. This map is given by a precise formula, using techniques from the theory of symplectic reduction and symplectic implosion. We then show, using the Vinberg monoid, that the gradient-Hamiltonian flow for a horospherical degeneration of an algebraic variety gives rise to this contraction from a general fiber to the special fiber. We apply this construction to branching problems in representation theory, and finally we show how the Gel'fand-Tsetlin integrable system can be understood to arise this way.

show abstract

“…The map Ψ generates a Hamiltonian action of T n ˆ¨¨¨ˆT a on a open, dense subset of O λ pn`1q , with respect to the canonical Kostant-Kirillov-Souriau symplectic structure on O λ pn`1q [GS83a]. It was shown in a more general setting by the second author that the dense subset where Ψ generates a torus action is connected and the fibers of Ψ are all connected [Lane18].…”

Section: The General Casementioning

confidence: 99%

The topology of Gelfand-Zeitlin fibers

Carlson¹,

Lane²

2021

Preprint

View full text Add to dashboard Cite

We prove several new results about the topology of fibers of Gelfand-Zeitlin systems on unitary and orthogonal coadjoint orbits. First, we provide a simplified description of their diffeomorphism types as balanced products, recovering results of Bouloc-Miranda-Zung as special cases. Second, we compute their cohomology rings. Finally, we complete the computation of their first three homotopy groups (the first and second homotopy groups were described by Cho-Kim-Oh in the unitary case). Our results build on Cho-Kim-Oh's description of Gelfand-Zeitlin fibers as total spaces of certain towers of fiber bundles. Our description of the diffeomorphism type, cohomology ring, and homotopy groups can be read in a straightforward manner from the combinatorics of the associated Gelfand-Zeitlin pattern.

show abstract

Convexity and Thimm’s Trick

Cited by 9 publications

References 40 publications

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

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

Contraction of Hamiltonian $K$-Spaces

The topology of Gelfand-Zeitlin fibers

Contact Info

Product

Resources

About