ORBITS OF THE SYMPLECTIC GROUP ON THE COMPLETE FLAG VARIETY OF A SYMPLECTIC SPACE

1995 ◽  
Vol 44 (1) ◽  
pp. 143-164
Author(s):  
A S Shmelëv
Keyword(s):  
Symplectic Group ◽  
Flag Variety ◽  
Symplectic Space ◽  
Complete Flag

SYMPLECTIC SPACE OR ORTHOGONAL SPACE OF n QUBITS

2011 ◽  
Vol 09 (06) ◽  
pp. 1449-1457
Author(s):  
JIAN-WEI XU
Keyword(s):  
Hilbert Space ◽  
Orthogonal Group ◽  
Symplectic Group ◽  
Orthonormal Basis ◽  
Mathematical Structure ◽  
Symplectic Space ◽  
Orthogonal Space ◽  
Spin Flip

In Hilbert space of n qubits, we introduce symplectic space (n odd) or orthogonal space (n even) via the spin-flip operator. Under this mathematical structure we discuss some properties of n qubits, including homomorphically mapping local operations of n qubits into symplectic group or orthogonal group, and proving that the generalized "magic basis" is just the biorthonormal basis (i.e. the orthonormal basis of both Hilbert space and the orthogonal space). Finally, a demonstrated example is given to discuss the application in physics of this mathematical structure.

scholarly journals Twisted gamma filtration and algebras with orthogonal involution

2014 ◽  
Vol 12 (3) ◽  
Author(s):  
Caroline Junkins
Keyword(s):  
Algebraic Group ◽  
Grothendieck Group ◽  
Linear Algebraic Group ◽  
Flag Variety ◽  
Flag Varieties ◽  
Torsion Element ◽  
Complete Flag ◽  
Gamma Filtration

AbstractFor the Grothendieck group of a split simple linear algebraic group, the twisted γ-filtration provides a useful tool for constructing torsion elements in -rings of twisted flag varieties. In this paper, we construct a non-trivial torsion element in the γ-ring of a complete flag variety twisted by means of a PGO-torsor. This generalizes the construction in the HSpin case previously obtained by Zainoulline.

scholarly journals 0-Hecke algebra actions on coinvariants and flags

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Jia Huang
Keyword(s):  
Generating Functions ◽  
Hecke Algebra ◽  
Flag Variety ◽  
Major Index ◽  
Inversion Number ◽  
International Audience ◽  
Descent Set ◽  
Complete Flag ◽  
Coinvariant Algebra

International audience By investigating the action of the 0-Hecke algebra on the coinvariant algebra and the complete flag variety, we interpret generating functions counting the permutations with fixed inverse descent set by their inversion number and major index. En étudiant l'action de l'algèbre de 0-Hecke sur l'algèbre coinvariante et la variété de drapeaux complète, nous interprétons les fonctions génératrices qui comptent les permutations avec un ensemble inverse de descentes fixé, selon leur nombre d'inversions et leur "major index''.

Symmetric Genuine Spherical Whittaker Functions on

2015 ◽  
Vol 67 (1) ◽  
pp. 214-240 ◽  
Author(s):  
Dani Szpruch
Keyword(s):  
Symplectic Group ◽  
Series Representation ◽  
Principal Series ◽  
Uniqueness Result ◽  
Symplectic Space ◽  
Whittaker Functions ◽  
Series Representations ◽  
Spanning Set ◽  
Double Covers ◽  
Principal Series Representations

AbstractLet F be a p-adic field of odd residual characteristic. Let and be the metaplectic double covers of the general symplectic group and the symplectic group attached to the 2n dimensional symplectic space over F, respectively. Let σ be a genuine, possibly reducible, unramified principal series representation of . In these notes we give an explicit formula for a spanning set for the space of Spherical Whittaker functions attached to σ. For odd n, and generically for even n, this spanning set is a basis. The significant property of this set is that each of its elements is unchanged under the action of the Weyl group of . If n is odd, then each element in the set has an equivariant property that generalizes a uniqueness result proved by Gelbart, Howe, and Piatetski-Shapiro.Using this symmetric set, we construct a family of reducible genuine unramified principal series representations that have more then one generic constituent. This family contains all the reducible genuine unramified principal series representations induced from a unitary data and exists only for n even.

scholarly journals A Divided Difference Operator

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Nicholas Teff
Keyword(s):  
Difference Operator ◽  
Divided Difference ◽  
Root Systems ◽  
Bruhat Order ◽  
Flag Variety ◽  
International Audience ◽  
Complete Flag ◽  
Special Case

International audience We construct a divided difference operator using GKM theory. This generalizes the classical divided difference operator for the cohomology of the complete flag variety. This construction proves a special case of a recent conjecture of Shareshian and Wachs. Our methods are entirely combinatorial and algebraic, and rely heavily on the combinatorics of root systems and Bruhat order.

scholarly journals Manifolds of Isospectral Matrices and Hessenberg Varieties

2020 ◽  
Author(s):  
Anton Ayzenberg ◽  
Victor Buchstaber
Keyword(s):  
Smooth Manifold ◽  
Flag Variety ◽  
Natural Action ◽  
Cohomology Rings ◽  
Compact Torus ◽  
Hessenberg Varieties ◽  
Toda Flow ◽  
Odd Degree ◽  
Complete Flag ◽  
The Given

Abstract We consider the space $X_h$ of Hermitian matrices having staircase form and the given simple spectrum. There is a natural action of a compact torus on this space. Using generalized Toda flow, we show that $X_h$ is a smooth manifold and its smooth type is independent of the spectrum. Morse theory is then used to show the vanishing of odd degree cohomology, so that $X_h$ is an equivariantly formal manifold. The equivariant and ordinary cohomology rings of $X_h$ are described using GKM theory. The main goal of this paper is to show the connection between the manifolds $X_h$ and regular semisimple Hessenberg varieties well known in algebraic geometry. Both spaces $X_h$ and Hessenberg varieties form wonderful families of submanifolds in the complete flag variety. There is a certain symmetry between these families, which can be generalized to other submanifolds of the flag variety.

scholarly journals Families of abelian surfaces with real multiplication over Hilbert modular surfaces

1982 ◽  
Vol 88 ◽  
pp. 17-53 ◽  
Author(s):  
G. van der Geer ◽  
K. Ueno
Keyword(s):  
Endomorphism Ring ◽  
Symplectic Group ◽  
Abelian Surface ◽  
Quadratic Relation ◽  
Holomorphic Map ◽  
Real Multiplication ◽  
Compact Complex Surfaces ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular ◽  
Abelian Functions

Around the beginning of this century G. Humbert ([9]) made a detailed study of the properties of compact complex surfaces which can be parametrized by singular abelian functions. A surface parametrized by singular abelian functions is the image under a holomorphic map of a singular abelian surface (i.e. an abelian surface whose endomorphism ring is larger than the ring of rational integers). Humbert showed that the periods of a singular abelian surface satisfy a quadratic relation with integral coefficients and he constructed an invariant D of such a relation with respect to the action of the integral symplectic group on the periods.

-ADIC -FUNCTIONS FOR ORDINARY FAMILIES ON SYMPLECTIC GROUPS

2019 ◽  
Vol 19 (4) ◽  
pp. 1287-1347 ◽  
Author(s):  
Zheng Liu
Keyword(s):  
Eisenstein Series ◽  
Symplectic Group ◽  
Symplectic Groups ◽  
Simple Strategy ◽  
Doubling Method

We construct the $p$-adic standard $L$-functions for ordinary families of Hecke eigensystems of the symplectic group $\operatorname{Sp}(2n)_{/\mathbb{Q}}$ using the doubling method. We explain a clear and simple strategy of choosing the local sections for the Siegel Eisenstein series on the doubling group $\operatorname{Sp}(4n)_{/\mathbb{Q}}$, which guarantees the nonvanishing of local zeta integrals and allows us to $p$-adically interpolate the restrictions of the Siegel Eisenstein series to $\operatorname{Sp}(2n)_{/\mathbb{Q}}\times \operatorname{Sp}(2n)_{/\mathbb{Q}}$.

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