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

Abstract We exhibit basic algebro-geometric results on the formal model of semi-infinite flag varieties and its Schubert varieties over an algebraically closed field ${\mathbb K}$ of characteristic $\neq 2$ from scratch. We show that the formal model of a semi-infinite flag variety admits a unique nice (ind-)scheme structure, its projective coordinate ring has a $\mathbb {Z}$ -model and it admits a Frobenius splitting compatible with the boundaries and opposite cells in positive characteristic. This establishes the normality of the Schubert varieties of the quasi-map space with a fixed degree (instead of their limits proved in [K, Math. Ann. 371 no.2 (2018)]) when $\mathsf {char}\, {\mathbb K} =0$ or $\gg 0$ , and the higher-cohomology vanishing of their nef line bundles in arbitrary characteristic $\neq 2$ . Some particular cases of these results play crucial roles in our proof [47] of a conjecture by Lam, Li, Mihalcea and Shimozono [60] that describes an isomorphism between affine and quantum K-groups of a flag manifold.

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Projections of Richardson varieties

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

10.1515/crelle-2012-0045 ◽

2014 ◽

Vol 2014 (687) ◽

Cited By ~ 6

Author(s):

Allen Knutson ◽

Thomas Lam ◽

David E Speyer

Keyword(s):

Flag Manifold ◽

Schubert Varieties ◽

Generalized Flag Manifold

Abstract.While the projections of Schubert varieties in a full generalized flag manifold

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Frobenius Splitting of Thick Flag Manifolds of Kac–Moody Algebras

International Mathematics Research Notices ◽

10.1093/imrn/rny174 ◽

2018 ◽

Vol 2020 (17) ◽

pp. 5401-5427 ◽

Cited By ~ 1

Author(s):

Syu Kato

Keyword(s):

Line Bundle ◽

Schubert Variety ◽

Flag Manifold ◽

Schubert Varieties ◽

Flag Manifolds ◽

Flag Varieties ◽

Moody Algebra ◽

Frobenius Splitting ◽

Projective Normality ◽

Plücker Relations

Abstract We explain that the Plücker relations provide the defining equations of the thick flag manifold associated to a Kac–Moody algebra. This naturally transplants the result of Kumar–Mathieu–Schwede about the Frobenius splitting of thin flag varieties to the thick case. As a consequence, we provide a description of the space of global sections of a line bundle of a thick Schubert variety as conjectured in Kashiwara–Shimozono [13]. This also yields the existence of a compatible basis of thick Demazure modules and the projective normality of the thick Schubert varieties.

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Topological representation of cloth state for robot manipulation

Autonomous Robots ◽

10.1007/s10514-021-09968-7 ◽

2021 ◽

Author(s):

Fabio Strazzeri ◽

Carme Torras

Keyword(s):

State Space ◽

Configuration Space ◽

Common Ground ◽

Low Complexity ◽

Flag Manifold ◽

Infinite Dimensional ◽

Articulated Objects ◽

Research Goal ◽

State Graphs ◽

Deformable Materials

AbstractForty years ago the notion of configuration space (C-space) revolutionised robot motion planning for rigid and articulated objects. Despite great progress, handling deformable materials has remained elusive because of their infinite-dimensional shape-state space. Finding low-complexity representations has become a pressing research goal. This work tries to make a tiny step in this direction by proposing a state representation for textiles relying on the C-space of some distinctive points. A stratification of the configuration space for n points in the cloth is derived from that of the flag manifold, and topological techniques to determine adjacencies in manipulation-centred state graphs are developed. Their algorithmic implementation permits obtaining cloth state–space representations of different granularities and tailored to particular purposes. An example of their usage to distinguish between cloth states having different manipulation affordances is provided. Suggestions on how the proposed state graphs can serve as a common ground to link the perception, planning and manipulation of textiles are also made.

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Polynomial identities related to special Schubert varieties

Applicable Algebra in Engineering Communication and Computing ◽

10.1007/s00200-021-00496-6 ◽

2021 ◽

Author(s):

Francesca Cioffi ◽

Davide Franco ◽

Carmine Sessa

Keyword(s):

Arbitrary Number ◽

Schubert Variety ◽

Decomposition Theorem ◽

Point Of View ◽

Explicit Description ◽

Intersection Cohomology ◽

Schubert Varieties ◽

Polynomial Identities ◽

Poincaré Polynomial

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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Additive kinematic formulas for flag area measures

Mathematische Annalen ◽

10.1007/s00208-021-02264-w ◽

2021 ◽

Author(s):

Judit Abardia-Evéquoz ◽

Andreas Bernig

Keyword(s):

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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