Cohomology of Spaltenstein varieties

Brundan, Jonathan; Ostrik, Victor

doi:10.1007/s00031-011-9149-2

Cited by 27 publications

(37 citation statements)

References 17 publications

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“…In the case G = GL n , every orbit O is equivariantly simply-connected and there is a natural G-equivariant semi-small resolution of singularities of O whose fibres admit affine pavings [BO11]. It follows that there exists a perverse parity sheaf E(O) with support O for any k as above.…”

Section: Toric Varietiesmentioning

confidence: 99%

Parity sheaves

Juteau

Mautner

Williamson

2014

J. Amer. Math. Soc.

187

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Given a stratified variety X X with strata satisfying a cohomological parity-vanishing condition, we define and show the uniqueness of “parity sheaves,” which are objects in the constructible derived category of sheaves with coefficients in an arbitrary field or complete discrete valuation ring. This construction depends on the choice of a parity function on the strata. If X X admits a resolution also satisfying a parity condition, then the direct image of the constant sheaf decomposes as a direct sum of parity sheaves, and the multiplicities of the indecomposable summands are encoded in certain refined intersection forms appearing in the work of de Cataldo and Migliorini. We give a criterion for the Decomposition Theorem to hold in the semi-small case. Our framework applies to many stratified varieties arising in representation theory such as generalised flag varieties, toric varieties, and nilpotent cones. Moreover, parity sheaves often correspond to interesting objects in representation theory. For example, on flag varieties we recover in a unified way several well-known complexes of sheaves. For one choice of parity function we obtain the indecomposable tilting perverse sheaves. For another, when using coefficients of characteristic zero, we recover the intersection cohomology sheaves and in arbitrary characteristic the special sheaves of Soergel, which are used by Fiebig in his proof of Lusztig’s conjecture.

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Section: Toric Varietiesmentioning

confidence: 99%

Parity sheaves

Juteau

Mautner

Williamson

2014

J. Amer. Math. Soc.

187

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“…Using this description together with Theorem 8.1, a proof of Conjecture 8.10 for g = sl n appears in [Wee, Theorem 8.3.7]. It is also possible to prove this result by showing explicitly that the B-algebra agrees in this case with the (straightforward) equivariant generalization of the Brundan-Ostrik presentation [BO11] of the cohomology of S3 varieties.…”

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confidence: 95%

Highest weights for truncated shifted Yangians and product monomial crystals

et al. 2019

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Truncated shifted Yangians are a family of algebras which are natural quantizations of slices in the affine Grassmannian. We study the highest weight representations of these algebras. In particular, we conjecture that the possible highest weights for these algebras are described by product monomial crystals, certain natural subcrystals of Nakajima's monomials. We prove this conjecture in type A. We also place our results in the context of symplectic duality and prove a conjecture of Hikita in this situation.

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“…In this action, the Chevalley generators of g act as certain trace maps associated to canonical adjunction maps between special translation functors that arise from tensoring with a g-module and its dual. In [15] it is explained that this action is closely connected to Ginzburg's geometric construction of representations of the general linear group [25]. Theorem 1.3 provides a context for understanding this surprising action by traces on category O.…”

Section: Quantum Group Categorificationsmentioning

confidence: 93%

Trace as an Alternative Decategorification Functor

et al. 2014

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Abstract. Categorification is a process of lifting structures to a higher categorical level. The original structure can then be recovered by means of the so-called "decategorification" functor. Algebras are typically categorified to additive categories with additional structure and decategorification is usually given by the (split) Grothendieck group. In this article we study an alternative decategorification functor given by the trace or the zeroth Hochschild-Mitchell homology. We show that this form of decategorification endows any 2-representation of the categorified quantum sln with an action of the current algebra U(sln[t]) on its center.

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Cohomology of Spaltenstein varieties

Abstract: We give a presentation for the cohomology algebra of the Spaltenstein variety of all partial flags annihilated by a fixed nilpotent matrix, generalizing the description of the cohomology algebra of the Springer fiber found by De Concini, Procesi and Tanisaki.Comment: 28 page

Cited by 27 publications

References 17 publications

Parity sheaves

Parity sheaves

Highest weights for truncated shifted Yangians and product monomial crystals

Trace as an Alternative Decategorification Functor

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