Unitary Subgroup Research Articles

Electric-field-enhanced second-harmonic domain contrast and nonreciprocity in a van der Waals antiferromagnet

Imaging antiferromagnetic 180° domains with actively controlled visibility is vital for both fundamental science and sophisticated applications. While optical second-harmonic generation (SHG) is a well-known technique for distinguishing such domains in non-centrosymmetric antiferromagnets, a general material-based strategy to control domain contrast remains elusive. Using van der Waals antiferromagnet MnPS3 as a proof of concept, we demonstrate the tuning of nonreciprocity-induced domain contrast in SHG through applying an in-plane electric field that transforms the magnetic point group to its unitary subgroup. The interference among intrinsic electric-dipole, magnetic-dipole, and field-induced electric-dipole transitions, each carrying distinct characters under space-inversion (P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal{P}}}$$\\end{document}) and time-reversal (T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal{T}}}$$\\end{document}) operations, enables large tuning of domain contrast and nonreciprocity in a broad spectral range. This strategy, generically applicable to systems characterized by PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal{P}}}{{\\mathcal{T}}}$$\\end{document}-symmetric magnetic groups with a polar unitary subgroup, offers a path to fast electrical modulation of nonlinear nonreciprocal photonic behaviors using antiferromagnets.

Deformation retraction of the group of strict contactomorphisms of the three-sphere to the unitary group

We prove that the group of strict contactomorphisms of the standard tight contact structure on the three-sphere deformation retracts to its unitary subgroup U(2).

Characterizations of normal cancellative monoids

<abstract><p>Normal cancellative monoids were introduced to explore the general structure of cancellative monoids, which are innovative and open up new possibilities. Specifically, we pointed out that the Green's relations in a cancellative monoid $ S $ are determined by its unitary subgroup $ U $ to a great extent. The specific composition of egg boxes in $ S $, derived from the general semigroup theory, can be settled by the subgroups of $ U $. We call a cancellative monoid normal when these subgroups are normal and characterize it as an NCM-system. This NCM-system was created in this article and can be obtained by combining a group and a condensed cancellative monoid. Furthermore, we introduced the concept of torsion extension and proved that a special kind of normal cancellative monoids can be constructed delicately by the outer automorphism groups of given groups and some simplified cancellative monoids.</p></abstract>

MSGCorep: A package for corepresentations of magnetic space groups

Motivated by easy access to complete corepresentation (corep) data of all the 1651 magnetic space groups (MSGs) in three-dimensional space, we have developed a Mathematica package MSGCorep to provide an offline database of coreps and various functions to manipulate them, based on our previous package SpaceGroupIrep. One can use the package MSGCorep to obtain the elements of any MSG and magnetic little group, to calculate the multiplication of group elements, to obtain the small coreps at any k-point and full coreps of any magnetic k-star for any MSG and show them in a user-friendly table form, to calculate and show the decomposition of direct products of full coreps between any two specified magnetic k-stars, and to determine the small coreps of energy bands. Both single-valued and double-valued coreps are supported. In addition, the 122 magnetic point groups (MPGs) and their coreps are also supported by this package. To the best of our knowledge, MSGCorep is the first package that is able to calculate the direct product of full coreps for any MSG and able to determine small coreps of energy bands for general purpose. In a word, the MSGCorep package is an offline database and tool set for MSGs, MPGs, and their coreps, and it is very useful to study the symmetries in magnetic (type-I, -III, and -IV MSGs) and nonmagnetic (type-II MSGs) materials. Program summaryProgram title:MSGCorepCPC Library link to program files:https://doi.org/10.17632/zt9r8pp8kr.1Developer's repository link:https://github.com/goodluck1982/MSGCorepLicensing provisions: GNU General Public Licence 3.0Programming language: WolframExternal routines/libraries used:SpaceGroupIrep (https://github.com/goodluck1982/SpaceGroupIrep)Nature of problem: The package MSGCorep provides offline database and tools for easy access to 1651 magnetic space groups, 122 magnetic point groups, and their corepresentations (coreps). MSGCorep is the first package that is able to calculate the direct product of full coreps for any magnetic space group and able to determine small coreps of energy bands for general purpose.Solution method: Coreps of a magnetic group are constructed from the representations of the maximal unitary subgroup of the magnetic group. Based on the complete representation data of 230 space groups provided by SpaceGroupIrep package, complete corep data of 1651 magnetic space groups are derived in MSGCorep package. A more convenient equation of induced corep, i.e. Eq. (2), is introduced to calculate the full coreps of all MSGs.

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].

Centre of Unitary Subgroup of Modular Group Algebra’s

We establish the structure of the centre of V∗(F2 k (M2n+1 )), 1 V∗(F2 k (M2n+1 × C2)) and V∗(F2 k ((M2n+1 × C2) × C2)) over a finite field of characteristics 2 where M2n+1 =< ψ, λ|ψ 2n = λ 2 = 1, λψ = ψ 2n+1λ > is the Modular group having order 2 n+1 and C2 is a cyclic group of order 2

ON UNITARY SUBGROUPS OF GROUP ALGEBRAS

Let $FG$ be the group algebra of a finite $p$-group $G$ over a
 finite field $F$ of characteristic $p$ and let $*$ be the
 classical involution of $FG$. The $*$-unitary subgroup of $FG$,
 denoted by $V_*(FG)$, is defined to be the set of all normalized
 units $u$ satisfying the property $u^*=u^{-1}$. In this paper we
 give a recursive method how to compute the order of the
 $*$-unitary subgroup for certain non-commutative group algebras.
 A variant of the modular isomorphism question of group algebras is
 also considered.

Unitary units of the group algebra of modular groups

Let [Formula: see text] be the group algebra of the modular group [Formula: see text] over a finite field [Formula: see text] of characteristic two. We calculate the order of the ∗-unitary subgroup of the group algebra [Formula: see text] and describe the structure of the ∗-unitary subgroup in the case when [Formula: see text].

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.

On groups with a strongly embedded unitary subgroup

The proper subgroup $B$ of the group $G$ is called {\it strongly embedded}, if $2\in\pi(B)$ and $2\notin\pi(B \cap B^g)$ for any element $g \in G \setminus B $ and, therefore, $ N_G(X) \leq B$ for any 2-subgroup $ X \leq B $. An element $a$ of a group $G$ is called {\it finite} if for all $ g\in G $ the subgroups $ \langle a, a^g \rangle $ are finite. In the paper, it is proved that the group with finite element of order $4$ and strongly embedded subgroup isomorphic to the Borel subgroup of $U_3(Q)$ over a locally finite field $Q$ of characteristic $2$ is locally finite and isomorphic to the group $U_3(Q)$. Keywords: A strongly embedded subgroup of a unitary type, subgroups of Borel, Cartan, involution, finite element.

The normal complement problem and the structure of the unitary subgroup

Let p be an odd prime and G be a finite split metabelian p-group of exponent p. In this article, we obtain a normal complement of G in where F is the field with p elements. Further, assume that where A is a finite abelian p-group and If F is any finite field of characteristic p, then we prove that G does not have a normal complement in and obtain the structure of the unitary subgroup Communicated by Sudarshan Kumar Sehgal

Normalized unit groups and their conjugacy classes

Let $$G = H \times A$$ be a finite 2-group, where H is a non-abelian group of order 8 and A is an elementary abelian 2-group. We obtain a normal complement of G in the normalized unit group V(FG) and in the unitary subgroup $$V_{*}(FG)$$ over the field F with 2 elements. Further, for a finite field F of characteristic 2, we derive class size of elements of V(FG). Moreover, we provide class representatives of $$V_{*}(FH)$$.

Deterministic twirling with low resources

Twirling operations, which average a quantum state with respect to a unitary subgroup, have become a frequently-employed tool in quantum information processing. We investigate the efficient implementation of twirling operations with minimal resources, without necessitating the ability to perform all possible unitary operations on the quantum system of interest. We present a general algebraic method allowing us to choose a set of - typically very few - unitary operators which, when applied randomly and repeatedly, produce the given twirling operation exponentially quickly. The method is applied to twirling operations for bipartite quantum systems with respect to the unitary group U(d)⊗U(d), an essential ingredient in entanglement distillation protocols. In particular, we provide a complete classification of sets of unitary operators capable of performing twirling on two qubits. Moreover, we construct a generic set containing at most three unitary operators achieving the twirling operation for a general two-qudit system.

CLASS LENGTH OF ELEMENTS OF GROUP IN THE NORMALIZED UNIT GROUP

Let F be a finite fi eld of characteristic p > 0. In this article, we obtain a relation between the class length of elements of a fi nite p-group G in the normalized unit group V(FG) and its unitary subgroup V*(FG), when p is an odd prime. We also provide the size of the conjugacy class of non-central elements of a group G in V(FG); where either G is any finite p-group with nilpotency class 2 or G is a p-group with nilpotency class 3 such that |G|<= p^5.

Projections with fixed difference: A Hopf-Rinow theorem

The set DA0, of pairs of orthogonal projections (P,Q) in generic position with fixed difference P−Q=A0, is shown to be a homogeneous smooth manifold: it is the quotient of the unitary group of the commutant {A0}′ divided by the unitary subgroup of the commutant {P0,Q0}′, where (P0,Q0) is any fixed pair in DA0. Endowed with a natural reductive structure (a linear connection) and the quotient Finsler metric of the operator norm, it behaves as a classic Riemannian space: any two pairs in DA0 are joined by a geodesic of minimal length. Given a base pair (P0,Q0), pairs in an open dense subset of DA0 can be joined to (P0,Q0) by a unique minimal geodesic.

Structure of the unitary subgroup of the group algebra 𝔽2n(QD)16

In this paper, we established the structure of unitary unit subgroup [Formula: see text] of the group algebra [Formula: see text], where [Formula: see text] is the Quasi-dihedral [D. S. Dummit and R. Foote, Abstract Algebra, 3rd edn. (Wiley, 2004), pp. 71–72] (Semi-Dihedral [B. Huppert, Endliche Gruppen (Springer, 1967), pp. 90–93]) group of order 16 and [Formula: see text] is any finite field of characteristic 2 with [Formula: see text] elements.

Free pairs of symmetric and unitary units in normal subgroups of a division ring

Let [Formula: see text] be the field of fractions of the group algebra [Formula: see text] of the Heisenberg group [Formula: see text], over the field [Formula: see text] of characteristic [Formula: see text]. We show that for some involutions of [Formula: see text] that are not induced from involutions of [Formula: see text], [Formula: see text] contains free symmetric and unitary pairs. We also give a general condition for a normal unitary subgroup of a division ring to contain a free group, and prove a generalization of Lewin’s Conjecture.

Units in modular group algebra

ABSTRACTLet 𝒰(FG) denotes the unit group of FG. In this article, we compute theorder of in terms of the order of 𝒰(FG) for an arbitrary finite group G, where is the cyclic group of order 2n and F is a finite field of characteristic 2. Further, if A is an elementary abelian 2-group, then we obtain structures of 𝒰(F(G×A)) and its unitary subgroup 𝒰∗(F(G×A)), where ∗ is the canonical involution of the group algebra F(G×A). Finally, we provide a set of generators of and 𝒰(FD4m).

On the order of unitary subgroup of the modular group algebra 𝔽2kD2N

Let 𝔽qD2Nbe the group algebra of D2N, the dihedral group of order 2N, over 𝔽q= GF (q). In this paper, we compute the order of the unitary subgroup of the group of units of 𝔽2kD2Nwith respect to the canonical involution ∗.

Units in $FD_{2p}$

In this note, we compute the order and provide the structure of the unit group $\mathcal{U}(FD_{2p^m})$ of the group algebra $FD_{2p^m}$, where $F$ is a finite field of characteristic 2 and $D_{2p^m}$ is the dihedral group of order $2p^m$ such that $p$ is an odd prime. Further, we obtain the structure of the unitary subgroup $\mathcal{U}_*(FD_{2p^m})$ with respect to canonical involution * and prove that it is a normal subgroup of the unit group $\mathcal{U}(FD_{2p^m})$.