Unitary Subgroups Research Articles

From orthosymplectic structure to super topological matter

Topological supermatter is given by ordinary topological matter constrained by supersymmetry or graded supergroups such as OSP(2N|2N). Using results on super oscillators and lattice QFT$_{d}$, we construct a super tight binding model on hypercubic super lattice with supercharge $ \boldsymbol{Q}=\sum_{\mathbf{k}}\hat{F}_{\mathbf{k}}.\boldsymbol{q}_{\mathbf{k}}.\hat{B}_{\mathbf{k}}$. We first show that the algebraic triplet $(\Omega ,G,J)$ of super oscillators can be derived from the $OSp(2N|2N)$ supergroup containing the symplectic $Sp(2N)$ and the orthogonal $SO(2N)$ as even subgroups. Then, we apply the obtained result on super oscillating matter to super bands and investigate its topological obstructions protected by TPC symmetries. We also give a classification of the Bose/Fermi coupling matrix $\boldsymbol{q}_{\mathbf{k}}$ in terms of subgroups of $OSp(2N|2N)$\ and show that there are $2P_{N}$ (partition of $N$) classes $\boldsymbol{q}_{\mathbf{k}}$ given by unitary subgroups of $U\left( 2\right) \times U\left( N\right) $. Other features are also given.

The order of the unitary subgroups of group algebras

Let [Formula: see text] be the group algebra of a finite [Formula: see text]-group [Formula: see text] over a finite field [Formula: see text] of positive characteristic [Formula: see text]. Let ⊛ be an involution of the algebra [Formula: see text] which is a linear extension of an anti-automorphism of the group [Formula: see text] to [Formula: see text]. If [Formula: see text] is an odd prime, then the order of the ⊛-unitary subgroup of [Formula: see text] is established. For the case [Formula: see text] we generalize a result obtained for finite abelian [Formula: see text]-groups. It is proved that the order of the ∗-unitary subgroup of [Formula: see text] of a non-abelian [Formula: see text]-group is always divisible by a number which depends only on the size of [Formula: see text], the order of [Formula: see text] and the number of elements of order two in [Formula: see text]. Moreover, we show that the order of the ∗-unitary subgroup of [Formula: see text] determines the order of the finite [Formula: see text]-group [Formula: see text].

On the isomorphism of unitary subgroups of noncommutative group algebras

Let FGFG be the group algebra of a finite pp-group GG over a field FF of characteristic pp. Let *\cd be an involution of the group algebra FGFG which arises form the group basis GG. The upper bound for the number of non-isomorphic *\cd-unitary subgroups is the number of conjugacy classes of the automorphism group GG with all the elements of order two. The upper bound is not always reached in the case when GG is an abelian group, but for non-abelian case the question is open. In this paper we present a non-abelian pp-group GG whose group algebra FGFG has sharply less number of non-isomorphic *\cd-unitary subgroups than the given upper bound.

The unitary subgroups of group algebras of a class of finite p-groups

Let p be a prime and let F be a finite field of characteristic p. Let FG denote the group algebra of the finite p-group G over the field F and let [Formula: see text] denote the group of normalized units in FG. The anti-automorphism [Formula: see text] of G extends linearly to an anti-automorphism [Formula: see text] of FG. An element [Formula: see text] is called unitary if [Formula: see text]. All unitary elements of [Formula: see text] form a subgroup which is denoted by [Formula: see text]. If p is odd, the order of [Formula: see text] is [Formula: see text]. However, to compute the order of [Formula: see text] still is open when [Formula: see text]. In this paper, the order of [Formula: see text] is computed when G is a nonabelian [Formula: see text]-group given by a central extension of the following form: [Formula: see text] and [Formula: see text], [Formula: see text]. Further, a conjecture is confirmed, namely, the order of [Formula: see text] can be divided by [Formula: see text], where [Formula: see text].

Geodesic neighborhoods in unitary orbits of self-adjoint operators of [formula omitted

In the present paper, we study the unitary orbit of a compact Hermitian diagonal operator with spectral multiplicity one under the action of the unitary group UK+C of the unitization of the compact operators K(H)+C, or equivalently, the quotient UK+C/UD(K+C). We relate this and the action of different unitary subgroups to describe metric geodesics (using a natural distance) which join end points. As a consequence we obtain a local Hopf-Rinow theorem. We also explore cases about the uniqueness of short curves and prove that there exist some of these that cannot be parameterized using minimal anti-Hermitian operators of K(H)+C.

Unitary Subgroups of Commutative Group Algebras of the Characteristic Two

Let FG be the group algebra of a finite 2-group G over a finite field F of characteristic two and let ⊛ be an involution that arises from G. The ⊛-unitary subgroup of FG is denoted by V⊛(FG) and defined as the set of all normalized units u satisfying the property u⊛ = u−1. We establish the order of V⊛(FG) for all involutions ⊛ arising from G, where G is a finite cyclic 2-group, and show that all ⊛-unitary subgroups of FG are not isomorphic.

The isomorphism problem of unitary subgroups of modular group algebras

The isomorphism problem of unitary subgroups of modular group algebras

2-Minimal subgroups of monomial, linear and unitary groups

This paper gives a detailed and explicit description of all the 2-minimal subgroups for the finite linear and unitary groups defined over fields of odd characteristic. Also the 2-minimal subgroups of all the subgroups of these groups which contain the special linear and special unitary subgroups are described. For a finite group G, a subgroup P of G is 2-minimal if B<P, where B=NG(S) for some Sylow 2-subgroup S of G, and B is contained in a unique maximal subgroup of P. The 2-minimal subgroups of certain monomial groups, which play an important role in this work, are also determined.

Unitary subgroups and orbits of compact self-adjoint operators

Fil: Bottazzi, Tamara Paula. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Saavedra 15. Instituto Argentino de Matematica Alberto Calderon; Argentina

Limit shapes for growing extreme characters of $U(\infty)$

We prove the existence of a limit shape and give its explicit description for certain probability distribution on signatures (or highest weights for unitary groups). The distributions have representation theoretic origin-they encode decomposition on irreducible characters of the restrictions of certain extreme characters of the infinite-dimensional unitary group $U(\infty)$ to growing finite-dimensional unitary subgroups $U(N)$. The characters of $U(\infty)$ are allowed to depend on $N$. In a special case, this describes the hydrodynamic behavior for a family of random growth models in $(2+1)$-dimensions with varied initial conditions.

Residues of standard intertwining operators on p-adic classical groups

Abstract We study two basic problems involved in the study of the standard intertwining operators attached to representations induced from irreducible unitary supercuspidal representations on maximal parabolic subgroups of p-adic classical groups. We give a formula to transfer the integrals over the unipotent radical to orbital integrals under the adjoint action of the Levi subgroup, and conclude that the residue of such standard intertwining operators at s = 0 ${s=0}$ is determined by compact orbital integrals.

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.

Entanglement verification using local unitary stabilizers

Local unitary stabilizer subgroups constitute powerful invariants for distinguishing various types of multipartite entanglement. In this paper, we show how stabilizers can be used as a basis for entanglement verification protocols on distributed quantum networks using minimal resources. As an example, we develop and analyze the performance of a protocol to verify membership in the space of Werner states, that is, multi-qubit states that are invariant under the action of any 1-qubit unitary applied to all the qubits.

Reduced Whitehead Groups and the Conjugacy Problem for Special Unitary Groups of Anisotropic Hermitian Forms

Let K/k be a separable field extension of degree 2, D be a finite-dimensional central division algebra over K with a K/k-involution τ, h be an Hermitian anisotropic form on a right D-vector space with respect to τ, and let U(h) be the unitary group of h. Then the reduced Whitehead group of its special linear subgroup is defined as follows: $$ \mathrm{SUK}_1^{\mathrm{an}}(h)={{{\mathrm{SU}(h)}} \left/ {{\left[ {U(h),U(h)} \right]}} \right.} $$ , where [U(h), U(h)] is the commutator subgroup of U(h). The first main result establishes a link between the above group and its analog SUK1(h) for the case of isotropic h (with respect to the same τ). Theorem. There exists a surjective homomorphism from $$ \mathrm{SUK}_1^{\mathrm{an}}(h) $$ to SUK1(h). Furthermore, we also give a solution of the conjugacy problem for special unitary subgroups of anisotropic Hermitian forms over quaternion division algebras as subgroups of their multiplicative groups. Bibliography: 32 titles.

Symmetric states: local unitary equivalence via stabilizers

We classify local unitary equivalence classes of symmetric states via a classification of their local unitary stabilizer subgroups. For states whose local unitary stabilizer groups have a positive number of continuous degrees of freedom, the classification is exhaustive. We show that local unitary stabilizer groups with no continuous degrees of freedom are isomorphic to finite subgroups of the rotation group $SO(3)$, and give examples of states with discrete stabilizers.

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).

Onlyn-Qubit Greenberger-Horne-Zeilinger States Are Undetermined by Their Reduced Density Matrices

The generalized n-qubit Greenberger-Horne-Zeilinger (GHZ) states and their local unitary equivalents are the only states of n qubits that are not uniquely determined among pure states by their reduced density matrices of n-1 qubits. Thus, among pure states, the generalized GHZ states are the only ones containing information at the n-party level. We point out a connection between local unitary stabilizer subgroups and the property of being determined by reduced density matrices.

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.

Invertible preservers and algebraic groups III: preservers of unitary similarity (congruence) invariants and overgroups of some unitary subgroups*

Let V = . We consider two groups of unitary operators on V, namely acting on V by conjugation, and acting on V via U:A→UAU 1. We study the overgroups of these unitary groups in GL(V). The results are then applied to classify the invertible linear maps on V and various subspaces of V which preserve some unitary similarity invariant or unitary congruence invariant property. This provides a unified proof for various existing results, and answers a number of open questions on linear preserver problems.

Consistency conditions for orientifolds and D-manifolds.

We study superstrings with orientifold projections and with generalized open string boundary conditions (D-branes). We find two types of consistency condition, one related to the algebra of Chan-Paton factors and the other to cancellation of divergences. One consequence is that the Dirichlet 5-branes of the Type I theory carry a symplectic gauge group, as required by string duality. As another application we study the Type I theory on a $K3$ $Z_2$ orbifold, finding a family of consistent theories with various unitary and symplectic subgroups of $U(16) \times U(16)$. We argue that the $K3$ orbifold with spin connection embedded in gauge connection corresponds to an interacting conformal field theory in the Type I theory.