Irreducible Representations Research Articles

Symmetry Reduction of States I

We develop a general theory of symmetry reduction of states on (possibly non-commutative) *-algebras that are equipped with a Poisson bracket and a Hamiltonian action of a commutative Lie algebra g . The key idea advocated for in this article is that the "correct" notion of positivity on a *-algebra A is not necessarily the algebraic one whose positive elements are the sums of Hermitian squares a^* a with a \in A , but can be a more general one that depends on the example at hand, like pointwise positivity on *-algebras of functions or positivity in a representation as operators. The notion of states (normalized positive Hermitian linear functionals) on A thus depends on this choice of positivity on A , and the notion of positivity on the reduced algebra A_{red} should be such that states on A_{red} are obtained as reductions of certain states on A . We discuss three examples in detail: Reduction of the *-algebra of smooth functions on a Poisson manifold M , which reproduces the coisotropic reduction of M ; reduction of the Weyl algebra with respect to translation symmetry; and reduction of the polynomial algebra with respect to a U(1)-action.

ON EXTERIOR POWERS OF REFLECTION REPRESENTATIONS

AbstractIn 1968, Steinberg [Endomorphisms of Linear Algebraic Groups, Memoirs of the American Mathematical Society, 80 (American Mathematical Society, Providence, RI, 1968)] proved a theorem stating that the exterior powers of an irreducible reflection representation of a Euclidean reflection group are again irreducible and pairwise nonisomorphic. We extend this result to a more general context where the inner product invariant under the group action may not necessarily exist.

Quantum Carroll/fracton particles

We classify and relate unitary irreducible representations (UIRs) of the Carroll and dipole groups, i.e., we define elementary quantum Carroll and fracton particles and establish a correspondence between them. Whenever possible, we express the UIRs in terms of fields on Carroll/Aristotle spacetime subject to their free field equations.We emphasise that free massive (or “electric”) Carroll and fracton quantum field theories are ultralocal field theories and highlight their peculiar and puzzling thermodynamic features. We also comment on subtle differences between massless and “magnetic” Carroll field theories and discuss the importance of Carroll and fractons symmetries for flat space holography.

Lagrangian formulation for free 6D infinite spin field

We construct a Lagrangian that describes the dynamics of a six-dimensional free infinite (continuous) spin field in 6D Minkowski space. The Lagrangian is formulated in the framework of the BRST approach to higher spin field theory and is based on a system of constraints defining an irreducible representation of the corresponding Poincaré group. The field realization of generators in the 6D Poincaré algebra and the second-, fourth-, and sixth-order Casimir operators are obtained in explicit form using additional spinor coordinates. Specific aspects of such a realization in six dimensions are discussed. We derive the conditions that determine the irreducible representation 6D infinite spin field and reformulate them as operators in the Fock space forming a first-class algebra in terms of commutators. These operators are used to construct the BRST charge and the corresponding Lagrangian. We prove that the conditions of the irreducible representation are reproduced as the consequence of the Lagrangian equations of motion, which finally provides the correctness of the results obtained.

A characterization of the L2-range of the Poisson transforms on a class of vector bundles over the quaternionic hyperbolic spaces

We study the L2-boundedness of the Poisson transforms associated to the homogeneous vector bundles Sp(n,1)×Sp(n)×Sp(1)Vτ over the quaternionic hyperbolic spaces Sp(n,1)/Sp(n)×Sp(1) associated with irreducible representations τ of Sp(n)×Sp(1) which are trivial on Sp(n). As a consequence, we describe the image of the section space L2(Sp(n,1)×Sp(n)×Sp(1)Vτ) under the generalized spectral projections associated to a family of eigensections of the Casimir operator.

Semi-invariant distribution vectors for p-adic unipotent groups

Let G be a unipotent algebraic group defined over a [Formula: see text]-adic field of characteristic zero. We denote by [Formula: see text] the set of rational points of G. It is a [Formula: see text]-adic Lie group with Lie algebra denoted by [Formula: see text]. Let [Formula: see text] be an irreducible unitary representation of [Formula: see text] in a Hilbert space [Formula: see text], [Formula: see text] be a linear form on [Formula: see text] and [Formula: see text] be a polarization at [Formula: see text]. We denote by [Formula: see text] a character of [Formula: see text] related to [Formula: see text]. The aim of this study is to give a precise description of the space of semi-invariant distribution vectors [Formula: see text] of [Formula: see text].

Integrality of the Betti moduli space

If in a given rank r r , there is an irreducible complex local system with torsion determinant and quasi-unipotent monodromies at infinity on a smooth quasi-projective variety, then for every prime number ℓ \ell , there is an absolutely irreducible ℓ \ell -adic local system of the same rank, with the same determinant and monodromies at infinity, up to semi-simplification. A finitely presented group is said to be weakly integral with respect to a torsion character and a rank r r if once there is an irreducible rank r r complex linear representation with determinant isomorphic to this torsion character, then for any ℓ \ell , there is an absolutely irreducible one of rank r r and determinant this given character, which is defined over Z ¯ ℓ \bar { \mathbb {Z}}_\ell . We prove that this property is a new obstruction for a finitely presented group to be the fundamental group of a smooth quasi-projective complex variety. The proofs rely on the arithmetic Langlands program via the existence of Deligne’s companions (L. Lafforgue, Drinfeld) and the geometric Langlands program via de Jong’s conjecture (Gaitsgory for ℓ ≥ 3 \ell \ge 3 ). We also define weakly arithmetic complex local systems and show they are Zariski dense in the Betti moduli. Finally we show that our method gives an arithmetic proof of the Corlette-T. Mochizuki theorem, proved using tame pure imaginary harmonic metrics, which shows the pull-back by a morphism between two smooth complex algebraic varieties of a semi-simple complex local system is semi-simple.

Representation ring of Levi subgroups versus cohomology ring of flag varieties III

For any reductive group G and a parabolic subgroup P with its Levi subgroup L, the first author [9] introduced a ring homomorphism ξλP:Repλ−polyC(L)→H⁎(G/P,C), where Repλ−polyC(L) is a certain subring of the complexified representation ring of L (depending upon the choice of an irreducible representation V(λ) of G with highest weight λ). In this paper we study this homomorphism for G=SO(2n) and its maximal parabolic subgroups Pn−k for any 2≤k≤n−1 (with the choice of V(λ) to be the defining representation V(ω1) in C2n). Thus, we obtain a C-algebra homomorphism ξn,kD:Repω1−polyC(SO(2k))→H⁎(OG(n−k,2n),C). We determine this homomorphism explicitly in the paper. We further analyze the behavior of ξn,kD when n tends to ∞ keeping k fixed and show that ξn,k becomes injective in the limit. We also determine explicitly (via some computer calculation) the homomorphism ξλP for all the exceptional groups G (with a specific ‘minimal’ choice of λ) and all their maximal parabolic subgroups except E8.

The second moment of Dirichlet twists of a text {GL}_{4} automorphic L-function

In this paper, we give an asymptotic formula for the second moment of Dirichlet twists of an automorphic L-function L(s, pi ) on the critical line averaged over characters and conductors, where pi denotes an irreducible tempered cuspidal automorphic representation of {textrm{GL}}_{4}({mathbb {A}}_{mathbb {Q}}) with unitary central character. We give some hybrid bound for the error term with respect to the size of conductors of Dirichlet characters and that of the automorphic representation.

Determinants of continuous visiting behavior to Palawan, Philippines: Integrating Uncertainty Reduction Theory and Expectation Confirmation Theory.

Palawan is a globally known island located in the Philippines. It has received several recognitions from a variety of international tourism organizations. The study aims to identify the factors affecting continuous visiting behavior to Palawan's travel accommodations. The study incorporated covariance-based structural equation modeling (CB-SEM) to apply ideologies of Uncertainty Reduction Theory (URT) and Expectation Confirmation Theory (ECT). 712 valid respondents answered an online questionnaire that was distributed to several social media platforms. Structural Equation Modeling (SEM) identified that interactive uncertainty significantly influenced perceived performance compared to passive uncertainty. Both the physical environment and attitude of employees significantly produced an impact on service experience and price acceptance. However, the physical environment negatively influenced price acceptance. Meanwhile, the attitude of employees was the sole exogenous variable that influenced price acceptance. Food and beverage didn't contribute a significant influence on service experience and price acceptance. Additionally, perceived performance and service experience contributed to positive confirmation. It was also revealed that confirmation was significant towards price acceptance, but was insignificant towards tourist satisfaction. Moreover, price acceptance significantly influenced tourist satisfaction. This study is the first comprehensive study that analyzed the visiting behavior in Palawan. Finally, integrating URT and ECT can be applied and extended, especially for analyzing the visiting behavior of different tourist destinations worldwide.

The Determinants and Prevalence of Drug Abuse by Gender in the Selected Competitive Sports at the Universities in Kenya

Purpose: The main objective of this study was to identify the determinants and prevalence of drug abuse by gender in selected competitive sports at the universities in Kenya.
 Methodology: The study used a mixed methods research design with an intention to capture and utilise both qualitative and quantitative data. The study sample comprised 300 respondents. Out of the 300 students, 173 (57.7%) were male and 127 (42.3%) were female players derived from six selected competitive sports at the universities in Kenya. The selected sports were badminton, tennis, basketball, volleyball, soccer and rugby. Stratified sampling followed by simple random sampling was used to select 10 universities proportionately (4 private and 6 public), the method was the most appropriate to capture universities with women rugby in addition to the five other selected sports. The study sought to investigate the prevalence of drug abuse due to factors such as psychosocial, medicinal and competition. SPSS computer version 20 was used to process data. The data was presented in form of Tables.
 Findings: The study found that more male players than females abused drugs, with majority being rugby players. Findings indicated that there were more males than females abusing drugs at approximately 89% and 11% respectively. It was concluded that more male than female athletes engaging in sports in universities in Kenya, abuse drugs. This could be attributed to the assumption that males tend to experiment more on most things than do their female counterparts.
 Unique Contribution to Theory, Practice and Policy: The Social Learning Theory, Self-Determination Theory and Harm Reduction Theory may be used to anchor future studies. It was recommended that various stakeholders that include the university management, NACADA and the security apparatus work together with an aim to sensitize and curb drug abuse menace among university athletes, general students’ community and the entire community bordering the universities.

Algebraic aspects of boundaries in the Kitaev quantum double model

We provide a systematic treatment of boundaries based on subgroups K ⊆ G for the Kitaev quantum double D(G) model in the bulk. The boundary sites are representations of a *-subalgebra Ξ ⊆ D(G) and we explicate its structure as a quasi-Hopf *-algebra dependent on a choice of transversal R. We provide decomposition formulae for irreducible representations of D(G) pulled back to Ξ. As an application of our treatment, we study patches with boundaries based on K = G horizontally and K = {e} vertically and show how these could be used in a quantum computer using the technique of lattice surgery. More abstractly, we also provide explicitly the monoidal equivalence of the category of Ξ-modules and the category of G-graded K-bimodules and use this to prove that different choices of R are related by Drinfeld cochain twists. Examples include Sn−1 ⊂ Sn and an example related to the octonions where Ξ is also a Hopf quasigroup.

Almost representations of algebras and quantization

abstract: We introduce the notion of almost representations of Lie algebras and quantum tori, and establish an Ulam-stability type phenomenon: every irreducible almost representation is close to a genuine irreducible representation. As an application, we prove that geometric quantizations of the two-dimensional sphere and the two-dimensional torus are conjugate in the semi-classical limit up to a small error.

A novel wake flow control method for drag reduction of a high-speed train with vortex generators installing on streamlined tail nose

The head/tail of a high-speed train has been designed in a streamlined way to achieve good aerodynamic performance, which leads to the flow separation point moving close to the tail nose tip. Therefore, the conventional way with some add-ons, i.e., the passive flow control, to suppress the flow separation in advance is not a good choice for this train wake flow improvement. Also, with the increase of the train speed, it is urgent to study new methods for the aerodynamic drag reduction of the train. The wake of the high-speed train is characterized by a pair of counter-rotating vortices, contributing to low surface pressure on the streamlined tail and posing a risk to the train operation. Thus, lowering the intensity of counter-rotating vortices and enhancing the surface pressure become a significantly potential drag reduction method. In the current study, a novel wake flow control method, named the vortex intensity reduction theory (VIRT), for the drag reduction of a high-speed train with vortex generators installing on the streamlined tail nose, was proposed to generate a pair of vortices with opposite rotating directions, expecting to weaken the wake vortices and have a higher-pressure distribution on the tail, as compared to the base case. The results show that with the installation of vortex generators (VGs), the train wake flow intensity is suppressed, and the influence region is reduced, resulting in the better train wake flow structures, as compared to the train without VGs. The VGs have significant impact on the aerodynamic performance of the tail car, while this effect is not evidently observed on the head and middle cars. The VGs contribute to the surface pressure increase on the streamlined tail, resulting into a reduction of pressure difference between the head and tail cars. As a result, a reduction of 5.11% in the aerodynamic drag and a reduction of 14.93% in the aerodynamic lift of the tail car are obtained, while for a three-car grouping train model, the reductions are about 2.23% and 72.66%, respectively. Therefore, the VIRT based on VGs proposed in this paper can effectively reduce the aerodynamic force of the tail car and alleviate the intensity of wake flow of the high-speed train, which will provide a newly potential drag reduction method of the next generation high-speed train.

Jahn–Teller distortion in Sr2FeO4: group-theoretical analysis and hybrid DFT calculations

We present theoretical justification for distorted Ruddlesden–Popper (RP) phases of the first-order by using hybrid density functional theory (DFT) calculations and group-theoretical analysis. We, thus, demonstrate the existence of the Jahn–Teller effect around an Fe^{4texttt {+}} ion in Sr_{2}FeO_{4}. On the calculation side, we have established a combination of Wu–Cohen (WC) exchange and Perdew-Wang (PW) correlation in a three-parameter functional WC3PW, giving the most accurate description of Sr_{2}FeO_{4} from the comparison of three hybrid DFT functionals. Self-consistently obtained Hartree–Fock exact exchange of 0.16 demonstrates consistent results with the experimental literature data. Importantly, we explain conditions for co-existing proper and pseudo-Jahn–Teller effects from the crystalline orbitals, symmetry-mode analysis and irreps products. Moreover, phonon frequency calculations support and confirm the results of symmetry-mode analysis. In particular, the symmetry-mode analysis identifies a dominating irreducible representation of the Jahn-Teller mode (X2+) and corresponding space group (SG) of ground state structure (SG Cmce model). Therefore, the usually suggested high-symmetry tetragonal crystal structure (SG I4/mmm model) is higher in energy by 121 meV/f.u. (equivalent to the Jahn-Teller stabilization energy) compared with the distorted low-symmetry structure (SG Cmce model). We also present diffraction patterns for the two crystal symmetries to discuss the differences. Therefore, our results shed light on the existence of low-symmetry RP phases and make possible direct comparisons with future experiments.

Fenomenologi Komunikasi Antarbudaya Anggota HIMMPAS

In everyday life, we always communicate with various kinds of people from various cultural, educational, economic and social backgrounds, this then makes us carry out intercultural communication. This also happened in one of the Student Activity Units (UKM) of Muhammadiyah University of Sidoarjo, namely HIMMPAS (Muhammadiyah Student Association of Sidoarjo Nature Lovers). This study aims to analyze the phenomenology of intercultural communication in the adaptation process of HIMMPAS members at Muhammadiyah University of Sidoarjo. The research method used in this research is qualitative with a phenomenological approach. The writing format is in the form of a case study by describing the subject's experiences with the epoch method. Data collection techniques in this study used observation, literacy studies, and interviews. Based on Uncertainty Reduction Theory, assumes that interaction is very important because the purpose of communication is to reduce uncertainty between strangers when meeting and having conversations. This study concludes that intercultural communication occurs among HIMMPAS members when members from Sidoarjo interact with members from outside Sidoarjo as well as between members from outside Sidoarjo and members who also come from outside Sidoarjo.

The Preference of Social Networking Sites and Uncertainty Reduction Strategies Towards Information on COVID-19 Vaccination

At the end of 2019, the world showcased a lethal outbreak of COVID-19, which was announced by the World Health Organization (WHO) as a pandemic in March 2020. As this outbreak has spread worldwide, including in Malaysia, several alternatives were undertaken by the government to curb this spread including introduce a vaccination program for the community. This program is believed to make most of the community worried or uncertain since it is still newly introduced. In reducing people's worries and uncertainties regarding COVID-19 vaccination, searching for information via social networking sites (SNS) will likely reduce their uncertainty. This study aims to determine the most preferred SNS used by people of middle age to reduce uncertainty regarding information on COVID-19 vaccination. This study also aimed to identify the factors that contribute to middle-aged people's preferences for SNS to reduce their uncertainty about COVID-19 vaccination, as well as the uncertainty reduction strategies (URSs) used by middle-aged people over preferred SNS to reduce their level of uncertainty about COVID-19 vaccination. Uncertainty Reduction Theory was used as the underpinning theory for this study. This study employed a qualitative approach in which an in-depth e-interview was conducted over a Google Meet platform. The results showed that Facebook is the most preferred SNS the informants use to reduce uncertainty regarding COVID-19 vaccination. This study also found information authority to be the most important factor for SNS preference. Finally, the study also discovered that the passive strategy is the most commonly employed method used by informants to reduce information uncertainties regarding COVID-19 vaccination. Keywords: COVID-19, vaccination, social networking sites, Uncertainty Reduction Theory, information.

On values of the Bessel function for generic representations of finite general linear groups

We find a recursive expression for the Bessel function of S. I. Gelfand for irreducible generic representations of GLn(Fq). We show that special values of the Bessel function can be realized as the coefficients of L-functions associated with exotic Kloosterman sums, and as traces of exterior powers of Katz's exotic Kloosterman sheaves. As an application, we show that certain polynomials, having special values of the Bessel function as their coefficients, have all of their roots lying on the unit circle. As another application, we show that special values of the Bessel function of the Shintani base change of an irreducible generic representation are related to special values of the Bessel function of the representation through Dickson polynomials.

Reducing Uncertainty Caused by Social Noise on Social Media Platform

Social noise and social entropy are two concepts that have been the subject of studies in the last two decades (Bruno, 2010; Matei et al.,2010; Alsaid et al., 2020; Alsaid & Pampapura, 2022; Pampapura et al., 2022). Studies that discussed misinformation, fake news, conspiracy theories, controversies, and disinformation also focused on certain aspects of social noise (Romer & Jamieson, 2020; Van Prooijen & Douglas, 2018). In this study, social noise refers to users intentionally or unintentionally participating in creating, disseminating, or spreading misinformation on social media. Recent studies have identified six characteristics or constructs of social noise. The six constructs identified by the recent studies include image curation, relationship management, cultural agency, conflict engagement, affiliation & politics, and norms & beliefs. In this study, we use the constructs to quantify and measure social noise using entropy and uncertainty reduction theory. The study focuses on two questions. How do we measure entropy as a degree of uncertainty caused by social noise? How does reducing uncertainty and social noise contribute to reducing the spread of misinformation? The primary analysis indicated that entropy as a measure of disorders of a tweet 1.5. This result is in line with the results produced by Son et al. (2019). Sentiment analysis results show that 66.6% of the tweets are subjective, while 33.4 % of the tweets are objective (facts). Topic modeling results also indicated the presence of keywords that is characteristics of social noise constructs in the dataset.

Energy bands in a three dimension simple cubic lattice of contact potential

In this work, we investigate energy bands in a three dimensional simple cubic lattice of contact potential. The energy bands in the first Brillouin Zone are obtained with Ewald’s summation method. In comparison with single point potential, the presence of lattice potential changes the existence condition of negative energy states near zero energy. It is found that the system always has negative energy states for an arbitrarily weak periodic potential. In addition, we prove that if an irreducible unitary representation is not a trivial representation of group of wave vector, the corresponding wave functions at lattice sites would be zero. With this theorem, the degeneracy of energy bands is explained with group theory. Furthermore, we find that there exists some energy bands which are not affected by the lattice potential. We call their corresponding eigenstates as dark states. The physical mechanism of the dark states is explained by explicitly constructing the standing wave-type Bloch wave functions.