scholarly journals Beilinson–Drinfeld Schubert varieties and global Demazure modules

2021 ◽  
Vol 9 ◽  
Author(s):  
Ilya Dumanski ◽  
Evgeny Feigin ◽  
Michael Finkelberg
Keyword(s):  
Line Bundle ◽  
Lie Algebra ◽  
Schubert Varieties ◽  
Affine Grassmannians ◽  
Demazure Modules ◽  
Determinant Line Bundle

Abstract We compute the spaces of sections of powers of the determinant line bundle on the spherical Schubert subvarieties of the Beilinson–Drinfeld affine Grassmannians. The answer is given in terms of global Demazure modules over the current Lie algebra.

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

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.

scholarly journals Determinants, Pfaffians and Quasi-Free Representations of the CAR Algebra

1998 ◽  
Vol 10 (05) ◽  
pp. 705-721 ◽  
Author(s):  
Mauro Spera ◽  
Tilmann Wurzbacher
Keyword(s):  
Hilbert Space ◽  
Line Bundle ◽  
Line Bundles ◽  
Complex Structures ◽  
Square Root ◽  
Purification Procedure ◽  
Algebraic Construction ◽  
Infinite Dimensional ◽  
Weil Type ◽  
Determinant Line Bundle

In this paper we apply the theory of quasi-free states of CAR algebras and Bogolubov automorphisms to give an alternative C*-algebraic construction of the Determinant and Pfaffian line bundles discussed by Pressley and Segal and by Borthwick. The basic property of the Pfaffian of being the holomorphic square root of the Determinant line bundle (after restriction from the Hilbert space Grassmannian to the Siegel manifold, or isotropic Grassmannian, consisting of all complex structures on an associated Hilbert space) is derived from a Fock–anti-Fock correspondence and an application of the Powers–Størmer purification procedure. A Borel–Weil type description of the infinite dimensional Spin c- representation is obtained, via a Shale–Stinespring implementation of Bogolubov transformations.

scholarly journals Cyclic Demazure Modules and Positroid Varieties

10.37236/8383 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Thomas Lam
Keyword(s):  
Canonical Basis ◽  
Schubert Varieties ◽  
Coordinate Ring ◽  
Demazure Module ◽  
Demazure Modules

A positroid variety is an intersection of cyclically rotated Grassmannian Schubert varieties.  Each graded piece of the homogeneous coordinate ring of a positroid variety is the intersection of cyclically rotated (rectangular) Demazure modules, which we call the cyclic Demazure module.  In this note, we show that the cyclic Demazure module has a canonical basis, and define the cyclic Demazure crystal.

scholarly journals Smoothness of Schubert varieties in twisted affine Grassmannians

2020 ◽  
Vol 169 (17) ◽  
pp. 3223-3260
Author(s):  
Thomas J. Haines ◽  
Timo Richarz
Keyword(s):  
Schubert Varieties ◽  
Affine Grassmannians

scholarly journals Remarks on Determinant Line Bundles, Chern-Simons Forms and Invariants

2002 ◽  
Vol 91 (1) ◽  
pp. 5 ◽  
Author(s):  
Johan L. Dupont ◽  
Flemming Lindblad Johansen
Keyword(s):  
Line Bundle ◽  
Line Bundles ◽  
Manifolds With Boundary ◽  
Principal Bundles ◽  
Determinant Line Bundles ◽  
Chern Simons ◽  
Determinant Line Bundle

We study generalized determinant line bundles for families of principal bundles and connections. We explore the connections of this line bundle and give conditions for the uniqueness of such. Furthermore we construct for families of bundles and connections over manifolds with boundary, a generalized Chern-Simons invariant as a section of a determinant line bundle.

scholarly journals DEMAZURE MODULES OF LEVEL TWO AND PRIME REPRESENTATIONS OF QUANTUM AFFINE 

2015 ◽  
Vol 17 (1) ◽  
pp. 75-105 ◽  
Author(s):  
Matheus Brito ◽  
Vyjayanthi Chari ◽  
Adriano Moura
Keyword(s):  
Lie Algebra ◽  
Cluster Algebras ◽  
Fusion Product ◽  
Parabolic Subalgebra ◽  
Highest Weight ◽  
Affine Lie Algebra ◽  
The Family ◽  
Demazure Module ◽  
Highest Weight Representations ◽  
Demazure Modules

We study the classical limit of a family of irreducible representations of the quantum affine algebra associated to $\mathfrak{sl}_{n+1}$. After a suitable twist, the limit is a module for $\mathfrak{sl}_{n+1}[t]$, i.e., for the maximal standard parabolic subalgebra of the affine Lie algebra. Our first result is about the family of prime representations introduced in Hernandez and Leclerc (Duke Math. J.154 (2010), 265–341; Symmetries, Integrable Systems and Representations, Springer Proceedings in Mathematics & Statitics, Volume 40, pp. 175–193 (2013)), in the context of a monoidal categorification of cluster algebras. We show that these representations specialize (after twisting) to $\mathfrak{sl}_{n+1}[t]$-stable prime Demazure modules in level-two integrable highest-weight representations of the classical affine Lie algebra. It was proved in Chari et al. (arXiv:1408.4090) that a stable Demazure module is isomorphic to the fusion product of stable prime Demazure modules. Our next result proves that such a fusion product is the limit of the tensor product of the corresponding irreducible prime representations of quantum affine $\mathfrak{sl}_{n+1}$.

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