The Grassmannian and Its Schubert Varieties

AbstractLet $$\mathcal S$$ S be a single condition Schubert variety with an arbitrary number of strata. Recently, an explicit description of the summands involved in the decomposition theorem applied to such a variety has been obtained in a paper of the second author. Starting from this result, we provide an explicit description of the Poincaré polynomial of the intersection cohomology of $$\mathcal S$$ S by means of the Poincaré polynomials of its strata, obtaining interesting polynomial identities relating Poincaré polynomials of several Grassmannians, both by a local and by a global point of view. We also present a symbolic study of a particular case of these identities.

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On a conjecture of Pappas and Rapoport about the standard local model for GL_d

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2020-0030 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Dinakar Muthiah ◽

Alex Weekes ◽

Oded Yacobi

Keyword(s):

Type A ◽

Positive Answer ◽

Local Model ◽

Schubert Varieties ◽

Shimura Varieties ◽

Local Models ◽

Full Generality ◽

Frobenius Splitting ◽

Affine Grassmannians ◽

Main Ideas

AbstractIn their study of local models of Shimura varieties for totally ramified extensions, Pappas and Rapoport posed a conjecture about the reducedness of a certain subscheme of {n\times n} matrices. We give a positive answer to their conjecture in full generality. Our main ideas follow naturally from two of our previous works. The first is our proof of a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman on the equations defining type A affine Grassmannians. The second is the work of the first two authors and Kamnitzer on affine Grassmannian slices and their reduced scheme structure. We also present a version of our argument that is almost completely elementary: the only non-elementary ingredient is the Frobenius splitting of Schubert varieties.

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On Schubert varieties in the flag manifold of Sl(n,?)

Mathematische Annalen ◽

10.1007/bf01450738 ◽

1987 ◽

Vol 276 (2) ◽

pp. 205-224 ◽

Cited By ~ 21

Author(s):

Kevin M. Ryan

Keyword(s):

Flag Manifold ◽

Schubert Varieties

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A Mathematica Program for the Degrees of Certain Schubert Varieties

Journal of Symbolic Computation ◽

10.1006/jsco.2001.0506 ◽

2002 ◽

Vol 33 (4) ◽

pp. 507-517 ◽

Cited By ~ 2

Author(s):

Xu an Zhao ◽

Haibao Duan

Keyword(s):

Schubert Varieties

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Poincaré polynomials of generic torus orbit closures in Schubert varieties

Topology, Geometry, and Dynamics - Contemporary Mathematics ◽

10.1090/conm/772/15490 ◽

2021 ◽

pp. 189-208

Author(s):

Eunjeong Lee ◽

Mikiya Masuda ◽

Seonjeong Park ◽

Jongbaek Song

Keyword(s):

Schubert Variety ◽

Flag Variety ◽

Schubert Varieties ◽

Orbit Closure ◽

Poincaré Polynomial ◽

Content Type ◽

Eulerian Polynomial ◽

Orbit Closures

The closure of a generic torus orbit in the flag variety G / B G/B of type A A is known to be a permutohedral variety, and its Poincaré polynomial agrees with the Eulerian polynomial. In this paper, we study the Poincaré polynomial of a generic torus orbit closure in a Schubert variety in G / B G/B . When the generic torus orbit closure in a Schubert variety is smooth, its Poincaré polynomial is known to agree with a certain generalization of the Eulerian polynomial. We extend this result to an arbitrary generic torus orbit closure which is not necessarily smooth.

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