Cayley Map for Symplectic Groups

2021 ◽  
Vol 22 ◽  
pp. 154-164
Author(s):  
Clementina D. Mladenova ◽  
Ivaïlo M. Mladenov
Keyword(s):  
Symplectic Groups ◽  
Symmetric Matrices ◽  
Exponential Map ◽  
Orthogonal Groups ◽  
Cayley Map ◽  
Four Blocks ◽  
Symplectic Matrices

Despite of their importance, the symplectic groups are not so popular like orthogonal ones as they deserve. The only explanation of this fact seems to be that their algebras can not be described so simply. While in the case of the orthogonal groups they are just the anti-symmetric matrices, those of the symplectic ones should be split in four blocks that have to be specified separately. It turns out however that in some sense they can be presented by the even dimensional symmetric matrices. Here, we present such a scheme and illustrate it in the lowest possible dimension via the Cayley map. Besides, it is proved that by means of the exponential map all such matrices generate genuine symplectic matrices.

Poles ofL-functions and theta liftings for orthogonal groups

2009 ◽  
Vol 8 (4) ◽  
pp. 693-741 ◽  
Author(s):  
David Ginzburg ◽  
Dihua Jiang ◽  
David Soudry
Keyword(s):  
Eisenstein Series ◽  
Orthogonal Group ◽  
Symplectic Groups ◽  
Orthogonal Groups ◽  
Metaplectic Groups ◽  
Automorphic Representations ◽  
Domain Of Holomorphy ◽  
First Occurrence ◽  
Cuspidal Automorphic Representations

AbstractIn this paper, we prove that the first occurrence of global theta liftings from any orthogonal group to either symplectic groups or metaplectic groups can be characterized completely in terms of the location of poles of certain Eisenstein series. This extends the work of Kudla and Rallis and the work of Moeglin to all orthogonal groups. As applications, we obtain results about basic structures of cuspidal automorphic representations and the domain of holomorphy of twisted standardL-functions.

scholarly journals A theorem on generation of finite orthogonal groups

1973 ◽  
Vol 16 (4) ◽  
pp. 495-506 ◽  
Author(s):  
W. J. Wong
Keyword(s):  
Finite Fields ◽  
Simple Groups ◽  
Symplectic Groups ◽  
Finite Simple Groups ◽  
Intermediate Result ◽  
Orthogonal Groups ◽  
Generators And Relations ◽  
Groups Of Lie Type

Presentation in terms of generators and relations for the classical finite simple groups of Lie type have been given by Steinberg and Curtis [2,4]. These presentations are useful in proving characterzation theorems for these groups, as in the author's work on the projective symplectic groups [5]. However, in some cases, the application is not quite instantaneous, and an intermediate result is needed to provide a presentation more suitable for the situation in hand. In this paper we prove such a result, for the orthogonal simple groups over finite fields of odd characteristic. In a subsequent article we shall use this to give a characterization of these groups in terms of the structure of the centralizer of an involution.

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.

Skew symmetric logarithms and geodesics on On(ℝ)

2018 ◽  
Vol 18 (4) ◽  
pp. 495-507 ◽  
Author(s):  
Alberto Dolcetti ◽  
Donato Pertici
Keyword(s):  
Symmetric Matrices ◽  
Exponential Map ◽  
Riemannian Structure ◽  
Orthogonal Matrices ◽  
Geometric Properties

Abstract We investigate the connections between the differential-geometric properties of the exponential map from the space of real skew symmetric matrices onto the group of real special orthogonal matrices and the manifold of real orthogonal matrices equipped with the Riemannian structure induced by the Frobenius metric.

scholarly journals Abelian 2-subgroups of finite symplectic groups in characteristic 2

1982 ◽  
Vol 33 (3) ◽  
pp. 345-350 ◽  
Author(s):  
M. J. J. Barry ◽  
W. J. Wong
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Vector Space ◽  
Symplectic Group ◽  
Preceding Paper ◽  
Symplectic Groups ◽  
Orthogonal Groups ◽  
Characteristic 2 ◽  
Finite Symplectic Groups

AbstractIf Sp(V) is the symplectic group of a vector space V over a finite field of characteristic p, and r is a positive integer, the abelian p-subgroups of largest order in Sp(V) whose fixed subspaces in V have dimension at least r were determined in the preceding paper, in the case p ≠ 2. Here we deal with the case p = 2. Our results also complete earlier work on the orthogonal groups.

scholarly journals Empirical Means on Pseudo-Orthogonal Groups

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 940
Author(s):  
Jing Wang ◽  
Huafei Sun ◽  
Simone Fiori
Keyword(s):  
Orthogonal Group ◽  
Geodesic Distance ◽  
Numerical Algorithms ◽  
Riemannian Metric ◽  
Frobenius Norm ◽  
Exponential Map ◽  
Orthogonal Groups ◽  
Orthogonal Matrices ◽  
Numerical Tests ◽  
Empirical Means

The present article studies the problem of computing empirical means on pseudo-orthogonal groups. To design numerical algorithms to compute empirical means, the pseudo-orthogonal group is endowed with a pseudo-Riemannian metric that affords the computation of the exponential map in closed forms. The distance between two pseudo-orthogonal matrices, which is an essential ingredient, is computed by both the Frobenius norm and the geodesic distance. The empirical-mean computation problem is solved via a pseudo-Riemannian-gradient-stepping algorithm. Several numerical tests are conducted to illustrate the numerical behavior of the devised algorithm.

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