Generators of infinite direct products of unitary groups

1977 ◽  
Vol 18 (11) ◽  
pp. 2166-2171 ◽  
Author(s):  
K. Kraus ◽  
L. Polley ◽  
G. Reents
Keyword(s):  
Unitary Groups ◽  
Direct Products

scholarly journals Singular fibres of the Gelfand–Cetlin system on MANI( n )*

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170423 ◽  
Author(s):  
D. Bouloc ◽  
E. Miranda ◽  
N.T. Zung
Keyword(s):  
Lie Groups ◽  
Homogeneous Spaces ◽  
Lagrangian Submanifolds ◽  
Unitary Groups ◽  
Direct Products ◽  
Compact Lie Groups ◽  
Lie Groupoids ◽  
Finite Dimensional ◽  
Adjoint Orbits ◽  
Free Actions

In this paper, we show that every singular fibre of the Gelfand–Cetlin system on co-adjoint orbits of unitary groups is a smooth isotropic submanifold which is diffeomorphic to a two-stage quotient of a compact Lie group by free actions of two other compact Lie groups. In many cases, these singular fibres can be shown to be homogeneous spaces or even diffeomorphic to compact Lie groups. We also give a combinatorial formula for computing the dimensions of all singular fibres, and give a detailed description of these singular fibres in many cases, including the so-called (multi-)diamond singularities. These (multi-)diamond singular fibres are degenerate for the Gelfand–Cetlin system, but they are Lagrangian submanifolds diffeomorphic to direct products of special unitary groups and tori. Our methods of study are based on different ideas involving complex ellipsoids, Lie groupoids and also general ideas coming from the theory of singularities of integrable Hamiltonian systems. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

scholarly journals Vertical Distribution Relations for Special Cycles on Unitary Shimura Varieties

2018 ◽  
Vol 2020 (13) ◽  
pp. 3902-3926
Author(s):  
Réda Boumasmoud ◽  
Ernest Hunter Brooks ◽  
Dimitar P Jetchev
Keyword(s):  
Vertical Distribution ◽  
Three Dimensional ◽  
Complex Multiplication ◽  
Horizontal Distribution ◽  
Unitary Groups ◽  
Shimura Varieties ◽  
Heegner Points ◽  
Universal Norm

Abstract We consider cycles on three-dimensional Shimura varieties attached to unitary groups, defined over extensions of a complex multiplication (CM) field $E$, which appear in the context of the conjectures of Gan et al. [6]. We establish a vertical distribution relation for these cycles over an anticyclotomic extension of $E$, complementing the horizontal distribution relation of [8], and use this to define a family of norm-compatible cycles over these fields, thus obtaining a universal norm construction similar to the Heegner $\Lambda $-module constructed from Heegner points.

scholarly journals The intersection graph of a finite simple group has diameter at most 5

2021 ◽  
Author(s):  
Saul D. Freedman
Keyword(s):  
Simple Group ◽  
Upper Bound ◽  
Finite Simple Group ◽  
Unitary Groups ◽  
Intersection Graph ◽  
Monster Group ◽  
Prime Dimension

AbstractLet G be a non-abelian finite simple group. In addition, let $$\Delta _G$$ Δ G be the intersection graph of G, whose vertices are the proper non-trivial subgroups of G, with distinct subgroups joined by an edge if and only if they intersect non-trivially. We prove that the diameter of $$\Delta _G$$ Δ G has a tight upper bound of 5, thereby resolving a question posed by Shen (Czechoslov Math J 60(4):945–950, 2010). Furthermore, a diameter of 5 is achieved only by the baby monster group and certain unitary groups of odd prime dimension.

Automorphisms of direct products of finite groups

2006 ◽  
Vol 86 (6) ◽  
pp. 481-489 ◽  
Author(s):  
J. N. S. Bidwell ◽  
M. J. Curran ◽  
D. J. McCaughan
Keyword(s):  
Finite Groups ◽  
Direct Products

scholarly journals Boundedness of direct products of torsion modules

1986 ◽  
Vol 39 ◽  
pp. 251-273 ◽  
Author(s):  
K.R. Goodearl ◽  
B. Zimmermann-Huisgen
Keyword(s):  
Direct Products ◽  
Torsion Modules
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