On the Saturation Conjecture for Spin(2<i>n</i>)

Kiers, Joshua

doi:10.1080/10586458.2018.1537866

Cited by 6 publications

(3 citation statements)

References 16 publications

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“…Here µ = a 1 ω 1 + a 2 ω 2 and µ = b 1 ω 1 + b 2 ω 2 + b 3 ω 3 are arbitrary dominant weights. The cohomology calculations were performed using Sage [29] using a modification of the main algorithm in [19]. These results agree with those of [23, §8.8], although they write their inequalities in a different basis.…”

Section: Case N =supporting

confidence: 64%

Extremal rays of the embedded subgroup saturation cone

Kiers¹

2022

Annales De l'Institut Fourier

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We examine the extremal rays of the cone of dominant weights (µ, µ) for groups G ⊆ G for which there exists N 0 such thatWe exhibit formulas for a class of rays ("type I") on any regular face. These rays are identified thanks to a generalization of Fulton's conjecture, which we prove. We verify that the remaining rays ("type II") on the face come from a smaller cone under a map whose formula is given. A procedure is given for finding the rays not on any regular face. This generalizes work of Belkale and Kiers on extremal rays for the saturated tensor cone; the specialization is given by G = G × G with the diagonal embedding of G. We include several examples to illustrate the formulas.Résumé. -Nous examinons les rayons extrémaux dans le cône des poids dominants des groupes G ⊆ G pour lesquels il existe N 0 tels queNous donnons des formules pour une classe de rayons (« type I ») sur une face régulière arbitraire. Ces rayons sont identifiés grace à une généralisation d'une conjecture de Fulton que nous démontrons. Nous vérifions que tous les autres rayons (« type 2 ») sur la face viennent d'un cône plus petit grâce à une application dont la formule est donnée. On décrit un processus pour trouver les rayons qui ne sont pas sur une face régulière. Ceci generalize les résultats de Belkale et Kiers sur les rayons extrémaux du cône tensoriel saturé; les résultats de ce papier sont démontrés ici quand on prend l'application diagonale de G dans G × G. Plusieurs exemples sont fournis.

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Section: Case N =supporting

confidence: 64%

Extremal rays of the embedded subgroup saturation cone

Kiers¹

2022

Annales De l'Institut Fourier

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“…There are examples in type A in [Bel17], due to Derksen-Weyman [DW11, Example 7.13] and Ressayre, of extremal rays for SLp8q and SLp9q respectively, which do not have this property, and give examples of extremal rays which are not type I on any face. There are similar examples which do not have this property for D 5 in [Kie18].…”

Section: Examplesmentioning

confidence: 99%

Extremal Rays in the Hermitian Eigenvalue Problem for Arbitrary Types

Belkale

Kiers

2019

Transformation Groups

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The Hermitian eigenvalue problem asks for the possible eigenvalues of a sum of Hermitian matrices given the eigenvalues of the summands. This is a problem about the Lie algebra of the maximal compact subgroup of G " SLpnq . There is a polyhedral cone (the "eigencone") determining the possible answers to the problem. These eigencones can be defined for arbitrary semisimple groups G, and also control the (suitably stabilized) problem of existence of non-zero invariants in tensor products of irreducible representations of G.We give a description of the extremal rays of the eigencones for arbitrary semisimple groups G by first observing that extremal rays lie on regular facets, and then classifying extremal rays on an arbitrary regular face. Explicit formulas are given for some extremal rays, which have an explicit geometric meaning as cycle classes of interesting loci, on an arbitrary regular face, and the remaining extremal rays on that face are understood by a geometric process we introduce, and explicate numerically, called induction from Levi subgroups. Several numerical examples are given. The main results, and methods, of this paper generalize [Bel17] which handled the case of G " SLpnq.

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“…does hold; this is because the Hermitian eigenvalue cones for H ∨ and H are isomorphic, as are those for G ∨ and G, see [KLM03, Theorem 1.8], and there is a map between the Hermitian eigenvalue cones for H and G since there is a compatible mapping of maximal compact subgroups, see [BK10]. Therefore implication (5.1) always holds when G is of type Kie19] by saturation. Here we note that Gr G,c( λ ′ ) = ∅ implies that λ ′ i is in the coroot lattice for G which equals the root lattice of G ∨ .…”

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confidence: 99%

A proof of the refined PRV conjecture via the cyclic convolution variety

Kiers¹

2019

Preprint

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In this brief note we illustrate the utility of the geometric Satake correspondence by employing the cyclic convolution variety to give a simple proof of the Parthasarathy-Ranga Rao-Varadarajan conjecture, along with Kumar's refinement. The proof involves recognizing certain MV-cycles as orbit closures of a group action, which we make explicit by unique characterization. In an appendix, joint with P. Belkale, we discuss how this work fits in a more general framework.

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On the Saturation Conjecture for Spin(2n)

Cited by 6 publications

References 16 publications

Extremal rays of the embedded subgroup saturation cone

Extremal rays of the embedded subgroup saturation cone

Extremal Rays in the Hermitian Eigenvalue Problem for Arbitrary Types

A proof of the refined PRV conjecture via the cyclic convolution variety

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