On the asymptotics of Kronecker coefficients

Manivel, Laurent

doi:10.1007/s10801-015-0614-1

Cited by 30 publications

(32 citation statements)

References 28 publications

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“…The first part of Theorem 5.5 permits us to recover the following result of Vallejo [28] and Manivel [15].…”

Section: Proofmentioning

confidence: 63%

“…Another interesting question is to produce examples of stable elements. In the case of Kronecker coefficients, Vallejo [28] and Manivel [15] introduced a notion of "additive matrix" that permits them to parametrize a large family of stable elements. In Section 5 we show that this notion can be transferred to the morphism case ρ (see Definition 5.1), and we compute the stretched coefficients associated to it.…”

Section: Introductionmentioning

confidence: 99%

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Stability property of multiplicities of group representations

Paradan

2019

Journal of Symplectic Geometry

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Let M be a compact complex manifold acted on by a compact Lie group G. Let L → M be a G-equivariant holomorphic line bundle that is assumed to be ample. Note that the G-action on L → M extends to the complex reductive group G C [9].In this context, we are interested in the family of G-modules Γ(M, L ⊗n ) formed by the holomorphic sections, and more particularly to the sequence H(n) := dim Γ(M, L ⊗n ) G , n ≥ 1. For any holomorphic G-complex vector bundle E → M , we consider also the sequenceOur main result, that we will detail in the next Section, can be summarized as follows : if the sequence H(n) is bounded, then the sequence H E (n) is bounded for any holomorphic G-complex vector bundle E and we can compute its value for large n. Stability resultSince the line bundle L is ample, there exists an Hermitian metric h on L such that the curvature Ω := i(∇ h ) 2 of its Chern connection ∇ h is a Kähler class : Ω is a symplectic form on M that is compatible with the complex structure. By an averaging process we can assume that the Gaction leaves the metric and connection invariant. Hence we have a moment map Φ : M → g * defined by Kostant's relationsHere L(X) is the Lie derivative on the sections of L, and X M (m) := d ds e −sX · m| s=0 is the vector field generated by X ∈ g.The [Q, R] = 0 Theorem of Meinrenken [16] and Meinrenken-Sjamaar [17] says that the moment map Φ gives a geometric interpretation of the sequence H(n). An important object here is the reduced space M 0 := Φ −1 (0)/G which is homeomorphic to the Mumford GIT quotient M/ /G C [11].

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“…The first part of Theorem 5.5 permits us to recover the following result of Vallejo [28] and Manivel [15].…”

Section: Proofmentioning

confidence: 63%

Section: Introductionmentioning

confidence: 99%

Stability property of multiplicities of group representations

Paradan

2019

Journal of Symplectic Geometry

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“…We have also verified our algorithm against computations made by different authors with various theoretical or computational aims (Hilbert series, stability, representations of the symmetric group, etc.). Here is a list that is likely far from being complete: [8,10,21,23,27,28,29,36,38,43]. In contrast to our method, some of these computations use directly the representation theory of the symmetric group (e.g., [23,36]).…”

Section: Type Inequalitymentioning

confidence: 99%

Computation of dilated Kronecker coefficients

Baldoni

Vergne

Walter

2018

Journal of Symbolic Computation

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Abstract. The computation of Kronecker coefficients is a challenging problem with a variety of applications. In this paper we present an approach based on methods from symplectic geometry and residue calculus. We outline a general algorithm for the problem and then illustrate its effectiveness in several interesting examples. Significantly, our algorithm does not only compute individual Kronecker coefficients, but also symbolic formulas that are valid on an entire polyhedral chamber. As a byproduct, we are able to compute several Hilbert series.

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“…Note that the extension encompasses all entangled states in the case N = 2, which are SLOCC-equivalent to the two-qubit W state. The conditions in (12) ensure that the support of p(λ|ψ) is compatible with the correspondence between partitions and marginal spectra, since replacing λ (i) by the spectra r (i) , the leftmost inequality in (12) gives the marginal spectral condition satisfied by all N -qubit states [17], while the rightmost inequality is the generalization of an additional spectral condition satisfied by W class states [18].…”

Section: Universality In the W-classmentioning

confidence: 99%

Universal and distortion-free entanglement concentration of multiqubit quantum states in the W class

Botero

Mejía

2018

Phys. Rev. A

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We propose a multipartite extension of Matsumoto and Hayashi's distortion-free entanglement concentration protocol, which takes n copies of a general multipartite state and, via local measurements, produces a maximally-entangled multipartite state between local spaces of dimensions ∼ 2 nE i , where Ei are the local entropies of the input state. However, the extended protocol is generally not universal in the sense that for the same measurement outcomes, the output state will still depend on the input state. Our main result is that when specialized to any state in the multiqubit W class, the protocol is also universal, so that as in the biparatite version, the output is a unique, maximally-entangled state for each given set of measurement outcomes. Our analysis brings to the forefront a new and interesting family of maximally-entangled multipartite states, which we term Kronecker states. A recurrence relation to obtain the coefficients of the W-class Kronecker states is also given.

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On the asymptotics of Kronecker coefficients

Cited by 30 publications

References 28 publications

Stability property of multiplicities of group representations

Stability property of multiplicities of group representations

Computation of dilated Kronecker coefficients

Universal and distortion-free entanglement concentration of multiqubit quantum states in the W class

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