Modern formulation; Grassmannians, flag varieties, schubert varieties

1998 ◽  
pp. 14-25
Author(s):  
William Fulton ◽  
Piotr Pragacz
Keyword(s):  
Schubert Varieties ◽  
Flag Varieties

scholarly journals Degenerate flag varieties and Schubert varieties: a characteristic free approach

2016 ◽  
Vol 284 (2) ◽  
pp. 283-308 ◽  
Author(s):  
Giovanni Cerulli Irelli ◽  
Martina Lanini ◽  
Peter Littelmann
Keyword(s):  
Schubert Varieties ◽  
Flag Varieties ◽  
Degenerate Flag Varieties

scholarly journals Total positivity for the Lagrangian Grassmannian

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Rachel Karpman
Keyword(s):  
Symplectic Form ◽  
Total Positivity ◽  
Combinatorial Structure ◽  
Schubert Varieties ◽  
Flag Varieties ◽  
Young Diagrams ◽  
Lagrangian Grassmannian ◽  
Planar Network ◽  
International Audience ◽  
Isotropic Subspaces

International audience The positroid decomposition of the Grassmannian refines the well-known Schubert decomposition, and has a rich combinatorial structure. There are a number of interesting combinatorial posets which index positroid varieties,just as Young diagrams index Schubert varieties. In addition, Postnikov’s boundary measurement map gives a family of parametrizations for each positroid variety. The domain of each parametrization is the space of edge weights of a weighted planar network. The positroid stratification of the Grassmannian provides an elementary example of Lusztig’s theory of total non negativity for partial flag varieties, and has remarkable applications to particle physics.We generalize the combinatorics of positroid varieties to the Lagrangian Grassmannian, the moduli space of maximal isotropic subspaces with respect to a symplectic form

scholarly journals Degenerate Flag Varieties of Type A and C are Schubert Varieties

2014 ◽  
Vol 2015 (15) ◽  
pp. 6353-6374 ◽  
Author(s):  
Giovanni Cerulli Irelli ◽  
Martina Lanini
Keyword(s):  
Type A ◽  
Schubert Varieties ◽  
Flag Varieties ◽  
Degenerate Flag Varieties

scholarly journals Frobenius splitting of Schubert varieties of semi-infinite flag manifolds

2021 ◽  
Vol 9 ◽  
Author(s):  
Syu Kato
Keyword(s):  
Positive Characteristic ◽  
Formal Model ◽  
Flag Manifold ◽  
Schubert Varieties ◽  
Flag Manifolds ◽  
Line Bundles ◽  
Closed Field ◽  
Flag Varieties ◽  
Fixed Degree ◽  
Frobenius Splitting

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.

scholarly journals Elliptic classes on Langlands dual flag varieties

2021 ◽  
pp. 2150014
Author(s):  
Richárd Rimányi ◽  
Andrzej Weber
Keyword(s):  
Mirror Symmetry ◽  
Homogeneous Spaces ◽  
Flag Variety ◽  
Schubert Calculus ◽  
Schubert Varieties ◽  
Characteristic Classes ◽  
Flag Varieties ◽  
K Theory

Characteristic classes of Schubert varieties can be used to study the geometry and the combinatorics of homogeneous spaces. We prove a relation between elliptic classes of Schubert varieties on a generalized full flag variety and those on its Langlands dual. This new symmetry is motivated by 3D mirror symmetry, and it is only revealed if Schubert calculus is elevated from cohomology or K theory to the elliptic level.

Projective normality of flag varieties and Schubert varieties

1985 ◽  
Vol 79 (2) ◽  
pp. 217-224 ◽  
Author(s):  
S. Ramanan ◽  
A. Ramanathan
Keyword(s):  
Schubert Varieties ◽  
Flag Varieties ◽  
Projective Normality

scholarly journals Frobenius Splitting of Thick Flag Manifolds of Kac–Moody Algebras

2018 ◽  
Vol 2020 (17) ◽  
pp. 5401-5427 ◽  
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.

Tropical flag varieties

2021 ◽  
Vol 384 ◽  
pp. 107695
Author(s):  
Madeline Brandt ◽  
Christopher Eur ◽  
Leon Zhang
Keyword(s):  
Flag Varieties

scholarly journals Polynomial identities related to special Schubert varieties

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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