Symmetric quivers, invariant theory, and saturation theorems for the classical groups

Sam, Steven V

doi:10.1016/j.aim.2011.10.009

Cited by 16 publications

(6 citation statements)

References 18 publications

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“…4 of [7]), and for B 2 via the expression (47). 2) One can use (29) (also formula (36) of [7]) that expresses V in terms of a finite number of multiplicities (LR coefficients) and of the constants rκ (defined in sect. 3) associated to g.…”

Section: The Many Ways To Compute the Volume Of A Bz Polytopementioning

confidence: 99%

On Horn’s Problem and Its Volume Function

Coquereaux

McSwiggen

Zuber

2019

Commun. Math. Phys.

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We consider an extended version of Horn's problem: given two orbits O α and O β of a linear representation of a compact Lie group, let A P O α , B P O β be independent and invariantly distributed random elements of the two orbits. The problem is to describe the probability distribution of the orbit of the sum A`B. We study in particular the familiar case of coadjoint orbits, and also the orbits of self-adjoint real, complex and quaternionic matrices under the conjugation actions of SOpnq, SUpnq and USppnq respectively. The probability density can be expressed in terms of a function that we call the volume function. In this paper, (i) we relate this function to the symplectic or Riemannian geometry of the orbits, depending on the case; (ii) we discuss its non-analyticities and possible vanishing; (iii) in the coadjoint case, we study its relation to tensor product multiplicities (generalized Littlewood-Richardson coefficients) and show that it computes the volume of a family of convex polytopes introduced by Berenstein and Zelevinsky. These considerations are illustrated by a detailed study of the volume function for the coadjoint orbits of B 2 " sop5q.

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Section: The Many Ways To Compute the Volume Of A Bz Polytopementioning

confidence: 99%

On Horn’s Problem and Its Volume Function

Coquereaux

McSwiggen

Zuber

2019

Commun. Math. Phys.

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“…Indeed, [KnTa99] adopts the former viewpoint, providing conjectural extensions to other Lie groups. Subsequent work includes [KaMi08,BeKu10,Ku10,Res10,Sa12]; see also the references therein.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Eigenvalues of Hermitian matrices and equivariant cohomology of Grassmannians

Anderson¹,

Richmond²,

Yong³

2013

Compositio Math.

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ABSTRACT. The saturation theorem of We further illustrate the common features between these two eigenvalue problems and their connection to Schubert calculus of Grassmannians. Our main result gives a Schubert calculus interpretation of Friedland's problem, via equivariant cohomology of Grassmannians. In particular, we prove a saturation theorem for this setting. Our arguments employ the aformentioned work together with .

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“…It was proved by Belkale-Kumar (2010) for the groups SO(2ℓ + 1) and Sp(2ℓ) by using geometric techniques. Sam (2012) proved it for SO(2ℓ) (and also for SO(2ℓ + 1) and Sp(2ℓ)) via the quiver approach (following the proof by Derksen-Weyman (2010) for G = SL(n)). (Observe that the general result of Kapovich-Millson gives a saturation factor of 4 in these cases.…”

Section: Theorem (14) For Any Connected Simple G D = Kmentioning

confidence: 93%

Additive Eigenvalue Problem (a survey), (With appendix by M. Kapovich)

Kumar

2013

Preprint

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Symmetric quivers, invariant theory, and saturation theorems for the classical groups

Cited by 16 publications

References 18 publications

On Horn’s Problem and Its Volume Function

On Horn’s Problem and Its Volume Function

Eigenvalues of Hermitian matrices and equivariant cohomology of Grassmannians

Additive Eigenvalue Problem (a survey), (With appendix by M. Kapovich)

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