scholarly journals A proof of Wahl's conjecture in the symplectic case

2013 ◽  
Vol 18 (1) ◽  
pp. 263-286
Author(s):  
Jesper Funch Thomsen
Keyword(s):  
Wahl’S Conjecture ◽  
Symplectic Case

scholarly journals Wahl’s conjecture for a minuscule G/P

2009 ◽  
Vol 119 (5) ◽  
pp. 571-592 ◽  
Author(s):  
J. Brown ◽  
V. Lakshmibai
Keyword(s):  
Wahl’S Conjecture

scholarly journals The Homology Class of a Poisson Transversal

2018 ◽  
Vol 2020 (10) ◽  
pp. 2952-2976
Author(s):  
Pedro Frejlich ◽  
Ioan Mărcuț
Keyword(s):  
Homology Class ◽  
Poisson Manifold ◽  
Poisson Manifolds ◽  
Poisson Structures ◽  
Symplectic Structures ◽  
Symplectic Case

Abstract This note is devoted to the study of the homology class of a compact Poisson transversal in a Poisson manifold. For specific classes of Poisson structures, such as unimodular Poisson structures and Poisson manifolds with closed leaves, we prove that all their compact Poisson transversals represent nontrivial homology classes, generalizing the symplectic case. We discuss several examples in which this property does not hold, as well as a weaker version of this property, which holds for log-symplectic structures. Finally, we extend our results to Dirac geometry.

scholarly journals Factorial Characters and Tokuyama's Identity for Classical Groups

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Angèle M. Hamel ◽  
Ronald C. King
Keyword(s):  
Recurrence Relations ◽  
Function Theory ◽  
Lattice Paths ◽  
Schur Function ◽  
Classical Groups ◽  
Orthogonal Groups ◽  
Schubert Polynomial ◽  
International Audience ◽  
Symplectic Case ◽  
Polynomial Theory

International audience In this paper we introduce factorial characters for the classical groups and derive a number of central results. Classically, the factorial Schur function plays a fundamental role in traditional symmetric function theory and also in Schubert polynomial theory. Here we develop a parallel theory for the classical groups, offering combinatorial definitions of the factorial characters for the symplectic and orthogonal groups, and further establish flagged factorial Jacobi-Trudi identities and factorial Tokuyama identities, providing proofs in the symplectic case. These identities are established by manipulating determinants through the use of certain recurrence relations and by using lattice paths.

scholarly journals Exact functors on perverse coherent sheaves

2015 ◽  
Vol 151 (9) ◽  
pp. 1688-1696
Author(s):  
Clemens Koppensteiner
Keyword(s):  
Symplectic Geometry ◽  
Coherent Sheaf ◽  
Alternative Definition ◽  
Coherent Sheaves ◽  
Constructible Sheaves ◽  
Definition Of ◽  
Symplectic Case

Inspired by symplectic geometry and a microlocal characterizations of perverse (constructible) sheaves we consider an alternative definition of perverse coherent sheaves. We show that a coherent sheaf is perverse if and only if $R{\rm\Gamma}_{Z}{\mathcal{F}}$ is concentrated in degree $0$ for special subvarieties $Z$ of $X$. These subvarieties $Z$ are analogs of Lagrangians in the symplectic case.

scholarly journals Abelian Unipotent Subgroups of Finite Unitary and Symplectic Groups

1982 ◽  
Vol 33 (3) ◽  
pp. 331-344 ◽  
Author(s):  
W. J. Wong
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Vector Space ◽  
Unitary Group ◽  
Symplectic Group ◽  
General Linear ◽  
Symplectic Groups ◽  
Orthogonal Groups ◽  
Abelian Subgroups ◽  
Symplectic Case

AbstractIf G is the unitary group U(V) or the symplectic group Sp(V) of a vector space V over a finite field of characteristic p, and r is a positive integer, we determine the abelian p-subgroups of largest order in G whose fixed subspaces in V have dimension at least r, with the restriction that we assume p ≠ 2 in the symplectic case. In particular, we determine the abelian subgroups of largest order in a Sylow p-subgroup of G. Our results complement earlier work on general linear and orthogonal groups.

scholarly journals Towards a Goldberg–Shahidi pairing for classical groups

2018 ◽  
Vol 30 (2) ◽  
pp. 347-384
Author(s):  
Arnab Mitra ◽  
Steven Spallone
Keyword(s):  
Adjoint Action ◽  
Bilinear Pairing ◽  
Vector Spaces ◽  
Unipotent Radical ◽  
Levi Subgroup ◽  
Intertwining Operators ◽  
Maximal Parabolic Subgroup ◽  
Induced Representations ◽  
Matrix Coefficients ◽  
Symplectic Case

AbstractLet{G^{1}}be an orthogonal, symplectic or unitary group over a local field and let{P=MN}be a maximal parabolic subgroup. Then the Levi subgroupMis the product of a group of the same type as{G^{1}}and a general linear group, acting on vector spacesXandW, respectively. In this paper we decompose the unipotent radicalNofPunder the adjoint action ofM, assuming{\dim W\leq\dim X}, excluding only the symplectic case with{\dim W}odd. The result is a Weyl-type integration formula forNwith applications to the theory of intertwining operators for parabolically induced representations of{G^{1}}. Namely, one obtains a bilinear pairing on matrix coefficients, in the spirit of Goldberg–Shahidi, which detects the presence of poles of these operators at 0.

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