Eigenvalue problem and a new product in cohomology of flag varieties

Belkale, Prakash; Kumar, Shrawan

doi:10.1007/s00222-006-0516-x

Cited by 85 publications

(214 citation statements)

References 31 publications

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“…In other words, where the expected dimension of s k=1 f i (X w k ) is zero for all i ∈ [r]. Equivalently, this numerical condition can be described by the notion of Levi-movability defined by Belkale and Kumar in [2]. Let E • := E a 1 ⊆ · · · ⊆ E ar ∈ Fℓ(a, n) denote the standard partial flag which is identified with the point eP ∈ SL n /P ≃ Fℓ(a, n) and let L be the Levi subgroup of P ⊆ SL n .…”

Section: Introductionmentioning

confidence: 99%

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A partial Horn recursion in the cohomology of flag varieties

Richmond

2008

J Algebr Comb

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Abstract. Horn recursion is a term used to describe when non-vanishing products of Schubert classes in the cohomology of complex flag varieties are characterized by inequalities parameterized by similar non-vanishing products in the cohomology of "smaller" flag varieties. We consider the type A partial flag variety and find that its cohomology exhibits a Horn recursion on a certain deformation of the cup product defined by Belkale and Kumar in [2]. We also show that if a product of Schubert classes is non-vanishing on this deformation, then the associated structure constant can be written in terms of structure constants coming from induced Grassmannians.

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Section: Introductionmentioning

confidence: 99%

“…The notion of L-movability defines a new product on the cohomology of Fℓ(a, n) which is a deformation of the usual product. For more details on this new product see [2]. Before we state the first main result of this paper, we need the following definition of induced permutations:…”

Section: Introductionmentioning

confidence: 99%

A partial Horn recursion in the cohomology of flag varieties

Richmond

2008

J Algebr Comb

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“…We consider on this group the Belkale-Kumar product 0 defined in [2]. The structure-coefficient of this product in this basis are either 0 or the structure-coefficient of the usual cup product.…”

Section: The Interior Of Ac G (B) Is Nonempty If and Only If No Nontrmentioning

confidence: 99%

Geometric Invariant Theory and Generalized Eigenvalue Problem II

Ressayre¹

2011

Ann. inst. Fourier

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Let G be a connected reductive subgroup of a complex connected reductive group Ĝ. Fix maximal tori and Borel subgroups of G and Ĝ. Consider the cone LR • ( Ĝ, G) generated by the pairs (ν, ν) of strictly dominant characters such that V ν is a submodule of V ν . The main result of this article is a bijective parametrisation of the faces of LR • ( Ĝ, G). We also explain when such a face is contained in another one.In way, we obtain results about the faces of the Dolgachev-Hu's Gample cone. We also apply our results to reprove known results about the moment polytopes.

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“…where δ is the Kronecker delta and [BK,Proposition 17] in the semisimple case, this product is associative (and of course commutative). The extension of this to the present affine Kac-Moody case requires a slight modification of the proof in loc cit.…”

Section: Theorem the Singular Cohomology Hmentioning

confidence: 99%

“…Moreover, there is a q-deformation of the cup product in H * (H/I, Z) such that the usual cup product corresponds to the value q = 1, whereas the new product 0 corresponds to the value q = 0 (cf. [BK,§6] Proof. From the definition of 0 , to prove the lemma, it suffices to show that for any u, v, w ∈ Aff 2 (W ) such that c w u,v = 0 in the decomposition (1) following Theorem 2.2, we have…”

Section: Theorem the Singular Cohomology Hmentioning

confidence: 99%

On the Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras

Kumar

2008

J. Amer. Math. Soc.

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We prove a part of the Cachazo-Douglas-Seiberg-Witten conjecture uniformly for any simple Lie algebra g \mathfrak {g} . The main ingredients in the proof are: Garland’s result on the Lie algebra cohomology of u ^ := g ⊗ t C [ t ] \hat {\mathfrak {u}} := \mathfrak {g}\otimes t\mathbb {C}[t] ; Kostant’s result on the ‘diagonal’ cohomolgy of u ^ \hat {\mathfrak {u}} and its connection with abelian ideals in a Borel subalgebra of g \mathfrak {g} ; and a certain deformation of the singular cohomology of the infinite Grassmannian introduced by Belkale-Kumar.

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Eigenvalue problem and a new product in cohomology of flag varieties

Cited by 85 publications

References 31 publications

A partial Horn recursion in the cohomology of flag varieties

A partial Horn recursion in the cohomology of flag varieties

Geometric Invariant Theory and Generalized Eigenvalue Problem II

On the Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras

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