Unitary Subgroup Research Articles

Decomposition of Unitary Linear Groups into Products of Free Factors

In this paper, we construct noncommutative algebras over a square-closed field such that the unitary linear groups over these algebras decompose to nontrivial free products. In particular, we give an example in which the elementary unitary subgroup belongs to one of these factors.

On the unitary units of the group algebra 𝔽2kQ16

In this note, the structure of unitary unit groups V*(𝔽2kQ16) is investigated, where Q16, is quaternion group of order 16, and 𝔽2kis any finite field of characteristic 2, with 2kelements. In particular, we give the center Z(V*(𝔽2kQ16)) of unitary units subgroup V*(𝔽2kQ16) of group algebra 𝔽2kQ16. The structure of the unitary unit subgroup V*(𝔽2kQ16) is described with the help of the Z(V*(𝔽2kQ16)).

Stable Groups Over Associative Rings with 1/2. A Description of Isomorphisms of the Stable Linear Groups

In this paper, we consider the stable unitary groups over associative rings with 1/2 and isomorphisms between them. We describe the action of the isomorphisms on the stable elementary unitary subgroup.

ON THE STRUCTURE OF THE UNITARY SUBGROUP OF THE GROUP ALGEBRA 𝔽2qD2n

We discuss the structure of the unitary subgroup V*(𝔽2qD2n) of the group algebra 𝔽2qD2n, where D2n= 〈x, y | x2n-1= y2= 1, xy = yx2n-1-1〉 is the dihedral group of order 2nand 𝔽2qis any finite field of characteristic 2, with 2qelements. We will prove that [Formula: see text], see Theorem 3.1.

Involutions and unitary subgroups in group algebras

Let FG be the group algebra of a finite group G over a field F of characteristic p. We give the maximal number of the non-isomorphic unitary subgroups with respect to the involutions of FG which arise from G. Furthermore, we characterize the group algebras with Hamiltonian unitary subgroup under the canonical involution, where G is a finite p-group and F is a finite field of characteristic p. Let FG denote the group algebra of a non-abelian group of order 8 over a finite field of characteristic two. We also describe the structure of the non-isomorphic unitary subgroups of FG linked to all the involutions which arise from G.

UNITS IN F2D2p

Let p be an odd prime, D2pbe the dihedral group of order 2p, and F2be the finite field with two elements. If * denotes the canonical involution of the group algebra F2D2p, then bicyclic units are unitary units. In this note, we investigate the structure of the group [Formula: see text], generated by the bicyclic units of the group algebra F2D2p. Further, we obtain the structure of the unit group [Formula: see text] and the unitary subgroup [Formula: see text], and we prove that both [Formula: see text] and [Formula: see text] are normal subgroups of [Formula: see text].

ON THE UNITARY UNITS OF THE GROUP ALGEBRA 𝔽2mM16

In this note, we have given the center Z(V*(𝔽2mM16)) of unitary units subgroup V*(𝔽2mM16) of group algebra 𝔽2mM16, where M16= 〈x, y | x8= y2= 1, xy = yx5〉 is the Modular group of order 16 and 𝔽2mis any finite field of characteristic 2, with 2melements. The structure of the unitary unit subgroup V*(𝔽2mM16) of the group algebra 𝔽2mM16, is also described, see Theorem 3.1.

The rational approximations of the unitary groups

This paper is to investigate the rational approximation of the unitary groups. Specially, based on the Household decomposition we prove that the rational unitary subgroup is dense in the complex unitary group. Moreover, its random approximate property is characterized by the natural Harr measure, which can be used to obtain random unitary matrix. Our simulation shows that these results may be applied to approximate quantum computations.

An Application of the Bruhat Decomposition to the Design of Full Diversity Unitary Space–Time Codes

A full diversity constellation, that is, a set of unitary matrices whose differences have nonzero determinant, is a design criterion for codes with good performance using differential unitary space-time modulation. Fixed-point free groups and the infinite group <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SU</i> (2) have been used to produce full diversity unitary group constellations. In this paper, we present a new Bruhat decomposition design for constructing full diversity unitary space-time constellations for any number of antennas. They are constructed from cosets of a unitary diagonal subgroup <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> , and our design has a particularly simple structure for the case where the number of transmitter antennas is prime. We also consider the extension of these constellation designs for an even number of transmitter antennas by replacing <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> with a Hamiltonian constellation. Some examples of proposed constellations for two to six transmitter antennas are given. Simulations show that our proposed constellations perform well in unknown Rayleigh fading channel.

On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices

Parametrization of 4 ◊ 4-matrices G of the complex linear group GL(4,C) in terms of four complex 4-vector parameters (k,m,n,l) is investigated. Additional restrictions separating some subgroups of GL(4,C) are given explicitly. In the given parametrization, the problem of inverting any 4◊4 matrix G is solved. Expression for determinant of any matrix G is found: detG = F(k,m,n,l). Unitarity conditions G + = G 1 have been formulated in the form of non-linear cubic algebraic equations including complex conjugation. Several simplest solutions of these unitarity equations have been found: three 2-parametric subgroups G1, G2, G3 - each of subgroups consists of two commuting Abelian unitary groups; 4-parametric unitary subgroup consisting of a product of a 3-parametric group isomorphic SU(2) and 1- parametric Abelian group. The Dirac basis of generators k, being of Gell-Mann type, substantially differs from the basis i used in the literature on SU(4) group, formulas relating them are found - they permit to separate SU(3) subgroup in SU(4). Special way to list 15 Dirac generators of GL(4,C) can be used { k} = { i j ( iV j = K L M)}, which permit to factorize SU(4) transformations according to S = e i~a~ e i ~b~ e ikK e ilL e imM , where two first factors commute with each other and are isomorphic to SU(2) group, the three last ones are 3-parametric groups, each of them consisting of three Abelian commuting unitary subgroups. Besides, the structure of fifteen Dirac matrices k permits to separate twenty 3-parametric subgroups in SU(4) isomorphic to SU(2); those subgroups might be used as bigger elementary blocks in constructing of a general transformation SU(4). It is shown how one can specify the present approach for the pseudounitary group SU(2,2) and SU(3,1).

Geometry of a two-qubit state and intertwining quaternionic conformal mapping under local unitary transformations

In this paper the geometry of two-qubit systems under a local unitary group SO(2) ⊗ SU(2) is discussed. It is shown that the quaternionic conformal map intertwines between this local unitary subgroup of Sp(2) and the quaternionic Möbius transformation which is rather a generalization of the results of Lee et al (2002 Quantum Inf. Process. 1 129).

A gaussian space of test functions

AbstractElsewhere we develop an aspect of analysis on the simplest space of type G/K when G = SL2(C) and K is the unitary subgroup, having to do with the analogue of Poisson's inversion formula and resulting additive zeta functions obtained by applying the Gauss transform. The present paper carries out systematically a number of foundational properties of gaussians on that space, which serve as test functions, give rise to explicit formulas, and lead immediately into the heat kernel, which is a gaussian. We prove approximation properties and the fact that the space of gaussians is dense in essentially anything space one wants. (© 2005 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

FREE SYMMETRIC AND UNITARY PAIRS IN DIVISION RINGS WITH INVOLUTION

Let D be a division ring with an involution and characteristic different from 2. Then, up to a few exceptions, D contains a pair of symmetric elements freely generating a free subgroup of its multiplicative group provided that (a) it is finite-dimensional and the center has a finite sufficiently large transcendence degree over the prime field, or (b) the center is uncountable, but not algebraically closed in D. Under conditions (a), if the involution is of the first kind, it is also shown that the unitary subgroup of the multiplicative group of D contains a free subgroup, with one exception. The methods developed are also used to exhibit free subgroups in the multiplicative group of a finite-dimensional division ring provided the center has a sufficiently large transcendence degree over its prime field.

Unitary subgroup space-time codes using bruhat decomposition and weyl groups

We propose a new class of unitary subgroups which is suitable to make differential unitary space-time codes for fast fading channels. These subgroups are derived from a double coset decomposition of unitary groups called Bruhat decomposition. We give some examples of the proposed unitary subgroup space-time codes for phase-shift keying (PSK), and display the performance for Rayleigh-fading channels. Finally, we describe how the new space-time codes are combined with coded modulation systems.

Residual Representations of Spacetime

Spacetime is modelled by binary relations—by the classes of the automorphisms \({\text{GL(}}\mathbb{C}^2 )\) of a complex two-dimensional vector space with respect to the definite unitary subgroup U(2). In extension of Feynman propagators for particle quantum fields representing only the tangent spacetime structure, global spacetime representations are given, formulated as residues using energy–momentum distributions with the invariants as singularities. The associated quantum fields are characterized by two invariant masses—for time and position—supplementing the one mass for the definite unitary particle sector with another mass for the indefinite unitary interaction sector without asymptotic particle interpretation.

On the order of the unitary subgroup of a modular group algebra

Let KG be a group algebra of a finite p-group G over a finite field Kof characteristic p. We compute the order of the unitary subgroup of the group of units when G is either an extraspecial 2-group or the central product of such a group with a cyclic group of order 4 or G has an abelian subgroup A of index 2 and an element b such that b inverts each element of A.

REPRESENTATIONS OF SPACETIME AS UNITARY OPERATION CLASSES; OR AGAINST THE MONOCULTURE OF PARTICLE FIELDS

Spacetime is modeled as a homogeneous manifoldgiven by the classes of unitary U(2) operations in thegeneral complex operations GL(\(\mathbb{C}^2 \)). The residual representations of thisnoncompact symmetric space of rank two are characterized by two continuousreal invariants, one invariant interpreted as a particlemass for a positive unitary subgroup and the second onefor an indefinite unitary subgroup related to nonparticle interpretable interaction ranges.Fields represent nonlinear spacetimeGL(\(\mathbb{C}^2 \))/U(2) by theirquantization and include necessarily nonparticlecontributions in the timelike part of their flat-space Feynman propagator.

Supercuspidal representations of GL(n) distinguished by a unitary subgroup

When E=F is a quadratic extension of p-adic elds, with p6 , and H 0 is a unitary similitude group in GL(n;E) ,i t is shown that for every irreducible supercuspidal representation of GL(n;E) of lowest level the space of H 0 -invariant linear forms has dimension at most one. The analogous fact for the corresponding unitary group H also holds, so long as n is odd or E=F is ramied. When n is even and E=F is unramied, the space of H-invariant linear forms on the space of may have dimension two.

Group theoretical approach to eigenvalue problems of symmetrical multiports with gyrotropic media

Eigenvalue problems of symmetrical multiports with gyrotropic media are considered. It is shown that the perturbation method known in quantum mechanics may be used for symmetrical structures with gyrotropic media in order to determine the influence of dc magnetic field on the eigenvalue degeneracy. In gyrotropic structures, an additional degeneracy may appear which is not predicted by the unitary subgroup. The eigenvalue equation for [S]-matrix in the case of magnetic groups of the third category and the problem of calculation of eigenvectors for transposed matrix are investigated. The theory is illustrated by some examples.

On the normalizers of the unitary subgroup in an integral group ring

In this paper, we investigate the properties of the normalizers of the unitary subgroup uf(ZG) in an integral group rings. One of our main results is Theorem 2.6 which character¬izes the second normalizer of the unitary subgroup. As a conse¬quence of this theorem, we prove that the second normalizer of uf(ZG) coincides with the first normalizer when G is a periodic group. Among other results, we give necessary and sufficient conditions for which the unitary subgroup is normal in the unit group when G is periodic and also characterize when all bicyclic units are nontrivial and elements of the normalizer of the unitary subgroup.