Irreducible Representations Research Articles

Preponderance of triaxial shapes in atomic nuclei predicted by the proxy-SU(3) symmetry

Abstract The proxy-SU(3) symmetry predicts, in a parameter-free way, based only on the Pauli principle and the short-range nature of the nucleon-nucleon interaction, non-vanishing values of the collective variable gamma almost everywhere across the nuclear chart. Substantial triaxiality with gamma between 15 and 45 degrees is proved to be expected along horizontal and vertical stripes on the nuclear chart, covering the nucleon numbers 22-26, 34-48, 74-80, 116-124, 172-182. Empirical support for these stripes is found by collecting all even-even nuclei for which the first two excited 2+ states are known, along with the B(E2)s connecting them, as well as the second 2+ state to the ground state. The stripes are related to regions in which oblate SU(3) irreducible representations appear, bearing similarity to the appearance of triaxiality within the SU(3)* dynamical symmetry of the interacting boson model-2. Detailed comparisons of the proxy-SU(3) predictions to the data and to predictions by state-of-the-art Monte Carlo shell model calculations for deformed N=94, 96, 98 isotones in the rare earth region show good overall agreement, with the exception of Z=70 and N=94, which correspond to fully symmetric proxy-SU(3) irreps, suggesting that the latter are an artifact of the method which can be amended by considering the influence of the neighboring irreps.

RATIONAL REPRESENTATIONS AND RATIONAL GROUP ALGEBRA OF VZ $\boldsymbol {p}$ -GROUPS

Abstract In this article, we study rational matrix representations of VZ p-groups (p is any prime). Using our findings on VZ p-groups, we explicitly obtain all inequivalent irreducible rational matrix representations of all p-groups of order $\leq p^4$ . Furthermore, we establish combinatorial formulae to determine the Wedderburn decompositions of rational group algebras for VZ p-groups and all p-groups of order $\leq p^4$ , ensuring simplicity in the process.

Decomposition of Thompson group representations arising from Cuntz algebras

Abstract For every integer n ≥ 2 n\geq 2 , we consider a family { π w } w ∈ { 0 , 1 , … , n − 1 } N \{\pi_{w}\}_{w\in\{0,1,\ldots,n-1\}^{\mathbb{N}}} of irreducible representations of the Cuntz algebra O n \mathcal{O}_{n} . All these representations (except one) are shown to be equivalent to those arising from the orbits of the interval map dynamical system ( I , f ) (I,f) with f ⁢ ( x ) = n ⁢ x ⁢ ( mod ⁢ 1 ) f(x)=nx\ (\mathrm{mod}\ 1) . We consider the embeddings V = V 2 ⊂ V n ⊂ O n V=V_{2}\subset V_{n}\subset\mathcal{O}_{n} of the Thompson group 𝑉 in the Higman–Thompson group V n V_{n} obtained by Birget and then from V n V_{n} into O n \mathcal{O}_{n} which was obtained independently by Birget and Nekrashevych. The restriction of π w \pi_{w} to V n V_{n} is still irreducible; however, the restriction of π w \pi_{w} to 𝑉 is no longer irreducible, and we obtain the corresponding irreducible decomposition in terms of quasi-regular representations.

Stealth dark matter spectrum using Laplacian Heaviside smearing and irreducible representations

We present nonperturbative lattice calculations in the quenched approximation of the low-lying meson and baryon spectrum of the SU(4) gauge theory with fundamental fermion constituents. This theory is one instance of stealth dark matter, a class of strongly coupled theories, where the lowest mass stable baryon is the dark matter candidate. This work constitutes the first milestone in the program to study stealth dark matter self-interactions. Here, we focus on reducing excited state contamination in the single-baryon channel by applying the Laplacian Heaviside method, as well as projecting our baryon operators onto the irreducible representations of the octahedral group. We compare our resulting spectrum to previous work involving Gaussian smeared nonprojected operators and find good agreement with reduced statistical uncertainties. We also present the spectrum of the low-lying odd-parity baryons for the first time. Published by the American Physical Society 2024

Riemann Zeta Function and Its Applications

Abstract. This paper focuses on the introduction and proof of the fundamental properties of (z), i.e. Riemann zeta function and explores its applications in algebra. We begin with a systematic derivation and proof of the basic characteristics of the zeta function. Following this, we examine its application in algebra, including the use of the Dirichlet L-function to prove Dirichlets theorem. Furthermore, we show the classical result for the subgroup growth rate of J-groups and the enumeration of n-dimensional irreducible representations of Heisenberg groups.

Construction and non-vanishing of a family of vector-valued Siegel Poincaré series

Using Poincaré series of K-finite matrix coefficients of integrable antiholomorphic discrete series representations of Sp2n(R), we construct a spanning set for the space Sρ(Γ) of Siegel cusp forms of weight ρ for Γ, where ρ is an irreducible polynomial representation of GLn(C) of highest weight ω∈Zn with ω1≥…≥ωn>2n, and Γ is a discrete subgroup of Sp2n(R) commensurable with Sp2n(Z). Moreover, using a variant of Muić's integral non-vanishing criterion for Poincaré series on unimodular locally compact Hausdorff groups, we prove a result on the non-vanishing of constructed Siegel Poincaré series.

Quantization in fibering polarizations, Mabuchi rays and geometric Peter–Weyl theorem

In this paper we use techniques of geometric quantization to give a geometric interpretation of the Peter–Weyl theorem. We present a novel approach to half-form corrected geometric quantization in a specific type of non-Kähler polarizations and study one important class of examples, namely cotangent bundles of compact connected Lie groups K. Our main results state that this canonically defined polarization occurs in the geodesic boundary of the space of K×K-invariant Kähler polarizations equipped with Mabuchi's metric, and that its half-form corrected quantization is isomorphic to the Kähler case. An important role is played by invariance of the limit polarization under a torus action.Unitary parallel transport on the bundle of quantum states along a specific Mabuchi geodesic, given by the coherent state transform of Hall, relates the non-commutative Fourier transform for K with the Borel–Weil description of irreducible representations of K.

Abstract irreducible representations of reductive algebraic groups with Borel-stable line

Abstract irreducible representations of reductive algebraic groups with Borel-stable line

Recent advances in branching problems of representations

How does an irreducible representation of a group behave when restricted to a subgroup? This is part of branching problems, which are one of the fundamental problems in representation theory, and also interact naturally with other fields of mathematics. This expository paper is an up-to-date account on some new directions in representation theory highlighting the branching problems for real reductive groups and their related topics ranging from global analysis of manifolds via group actions to the theory of discontinuous groups beyond the classical Riemannian setting. This article is an outgrowth of the invited lecture that the author delivered at the commemorative event for the 70th anniversary of the re-establishment of the Mathematical Society of Japan, and originally appeared in Japanese in Sugaku 71 (2019).

Symmetry Preservation in Hamiltonian Systems: Simulation and Learning

This work presents a general geometric framework for simulating and learning the dynamics of Hamiltonian systems that are invariant under a Lie group of transformations. This means that a group of symmetries is known to act on the system respecting its dynamics and, as a consequence, Noether’s theorem, conserved quantities are observed. We propose to simulate and learn the mappings of interest through the construction of G-invariant Lagrangian submanifolds, which are pivotal objects in symplectic geometry. A notable property of our constructions is that the simulated/learned dynamics also preserves the same conserved quantities as the original system, resulting in a more faithful surrogate of the original dynamics than non-symmetry aware methods, and in a more accurate predictor of non-observed trajectories. Furthermore, our setting is able to simulate/learn not only Hamiltonian flows, but any Lie group-equivariant symplectic transformation. Our designs leverage pivotal techniques and concepts in symplectic geometry and geometric mechanics: reduction theory, Noether’s theorem, Lagrangian submanifolds, momentum mappings, and coisotropic reduction among others. We also present methods to learn Poisson transformations while preserving the underlying geometry and how to endow non-geometric integrators with geometric properties. Thus, this work presents a novel attempt to harness the power of symplectic and Poisson geometry toward simulating and learning problems.

Layer-By-Layer Magnetic Ordering via Idle Spins and the Optical Signature of Jahn-Teller Cr2+ Ions in Sr2Cr(PO4)2.

Blue/turquoise crystals of Sr2CrII(PO4)2 with prismatic shape and edge-length of up to 1 mm were obtained by a vapor-phase moderated solid-state reaction at 1273 K in sealed silica tubes. Its crystal structure was solved and refined from a triply twinned ("trilling") crystal [Pbca (no. 61), Z = 12, a = 10.7064(6) Å, b = 9.2730(5) Å, c = 21.2720(7) Å, R1 = 0.038]. Sr2Cr(PO4)2 belongs to the small family of inorganic solids containing divalent chromium, where the rare Cr2+ ions are stabilized by the inductive effect of the phosphate groups. As expected from its d4 (S = 2) electronic configuration, the Jahn-Teller effect (JT) is prominent, leading for the two independent Cr2+ ions to square-pyramidal Cr(1)O4+1 and square-planar Cr(2)O4 coordination within a 3D chromium phosphate network [CrII2(PO4)4]8. Topologically, the Cr(1) and Cr(2) cations are arranged in separate alternating layers stacked along the c axis. In their respective layers, Cr(1) shows a gapped 2D topology and only weak interaction with the adjacent Cr(2) layers. However, below TN1 ∼11.3 K, Cr(1) orders antiferromagnetically into a noncollinear structure, leaving nearly paramagnetic Cr(2) idle spins, strongly frustrated by the Cr(1) moments of the next layers. On further cooling, below TN2 ∼3.6 K, the ordering of Cr(2) occurs via an additional magnetic irreducible representation, which splits the Cr(1) into Cr(1)a and Cr(1)b orbits, thus lifting the frustration on Cr(2). The corresponding P21ca.29.99 magnetic space group forces a crystal symmetry lowering, plausibly signed by a change of the magnetostrictive coefficient from positive to negative below TN2. The optical transitions observed for the JT d4 ions are in good agreement with our crystal picture from the DFT calculations. A detailed analysis within the angular overlap model explains the surprisingly different d orbital splitting by the ligand field for the chromophores Cr(1)O4+1 and Cr(2)O4.

Erratum: Non-admissible irreducible representations of 𝑝-adic 𝐺𝐿_{𝑛} in characteristic 𝑝

Erratum: Non-admissible irreducible representations of 𝑝-adic 𝐺𝐿_{𝑛} in characteristic 𝑝

Symmetry Reductions of the (1 + 1)-Dimensional Broer–Kaup System Using the Generalized Double Reduction Method

The generalized theory of the double reduction of systems of partial differential equations (PDEs) based on the association of conservation laws with Lie–Bäcklund symmetries is one of the most effective algorithms for performing symmetry reductions of PDEs. In this article, we apply the theory to a (1 + 1)-dimensional Broer–Kaup (BK) system, which is a pair of nonlinear PDEs that arise in the modeling of the propagation of long waves in shallow water. We find symmetries and construct six local conservation laws of the BK system arising from low-order multipliers. We establish associations between the Lie point symmetries and conservation laws and exploit the association to perform double reductions of the system, reducing it to first-order ordinary differential equations or algebraic equations. Our paper contributes to the broader understanding and application of the generalized double reduction method in the analysis of nonlinear PDEs.

The Relationship Between Misinformation And Social Noise And Their Impact On The Information Ecosystem

Social noise and social entropy are new concepts modeled after Shannon’s information and communication theory, in which the interference of noise between the sender and receiver is measured using entropy. Social noise in the context of social media plays a vital role in magnifying and spreading misinformation, which in turn impacts the overall information ecosystem. Ecosystems are made of interconnected and integrated parts that rely on one another to maintain balance and survive. Studies related to social noise and misinformation have shown that social noise can contribute significantly to spreading misinformation and potentially alter the original intended message (Alsaid & Pampapura, 2022); Alsaid et al. (2024). paper investigates methods of quantifying social noise using entropy to minimize the spread of misinformation on social media, particularly X. Using a combination of Uncertainty Reduction Theory (URT) and Social Entropy, data analysis was performed using one million tweets harvested from #Ukraine. Data analysis involved several methods: sentiment analyses, term association, network maps, and entropy computation. Results have shown a direct relationship between social noise and social entropy as a measure of uncertainty. Also, social noise and uncertainty decrease with the use of URLs and rich content. It is evident from the results that the entropy value is influenced by the accuracy of keywords identified using topic modeling as descriptive of social noise constructs. Semantic analysis of tweets can help improve the definition of social noise constructs, leading to enhanced and more accurate entropy calculation. Future studies may consider advanced machine learning and AI

Digitization and subduction of SU(N) gauge theories

The simulation of lattice gauge theories on quantum computers necessitates digitizing gauge fields. One approach involves substituting the continuous gauge group with a discrete subgroup, but the implications of this approximation still need to be clarified. To gain insights, we investigate the subduction of SU(2) and SU(3) to discrete crystal-like subgroups. Using classical lattice calculations, we show that subduction offers valuable information based on subduced direct sums, helping us identify additional terms to incorporate into the lattice action that can mitigate the effects of digitization. Furthermore, we compute the static potentials of all irreducible representations of Σ(360×3) at a fixed lattice spacing. Our results reveal a percent-level agreement with the Casimir scaling of SU(3) for irreducible representations that subduce to a single Σ(360×3) irreducible representation. This provides a diagnostic measure of approximation quality, as some irreducible representations closely match the expected results while others exhibit significant deviations. Published by the American Physical Society 2024

Service task compatibility for human staff vs. service robots: Roles of product type and brand level

This study explores when customers prefer service staff to service robots, or vice versa, based on uncertainty reduction theory. This study also investigates the moderating role of product type and brand level in the relationship between service agent type and experience satisfaction through perceived credibility and customer attitude. Two 2×2 between-subject experiments were conducted to test the moderated serial mediation models with product type in Study 1 and brand level in Study 2. Results show that customers tend to show higher experience satisfaction via perceived credibility and attitude toward the service agent type when they interact with service staff for inquiries of experiential products and upscale brands, compared to material products and casual brands. The study offers ways to maximize customer experience satisfaction by interactively and optimally deploying service agents and service robots for guest recommendation services, depending on the type of product and the level of brand.

Identification of optically active vibrational modes of columbite AB2O6 using the correlation method

In the modern era, the examination of molecular structure heavily relies on the application of infrared and Raman spectra within crystalline structures. These methodologies are essential in understanding the arrangement of atoms within molecules and the internal forces governing them. An essential aspect of this analysis involves identifying vibrational modes that can be detected optically. The correlation method is employed to establish rules for selecting these vibrational modes, both in crystals and molecules, through a systematic calculation process that aids in predicting their activity in Infrared (IR) and Raman spectra. The correlation method utilizes group theory to determine which vibrational modes are spectroscopically active in crystals. In our research, our aim is to employ this method to identify the irreducible representations and determine the IR and Raman active vibrational modes of orthorhombic AB2O6 compounds within the Pbcn space group. By conducting comprehensive group theory calculations, our objective is to elucidate the spectroscopic properties using the correlation method.

Ghost Distributions on Supersymmetric Spaces I: Koszul Induced Superspaces, Branching, and the Full Ghost Centre

AbstractGiven a Lie superalgebra $$\mathfrak {g}$$ g , Gorelik defined the anticentre $$\mathcal {A}$$ A of its enveloping algebra, which consists of certain elements that square to the center. We seek to generalize and enrich the anticentre to the context of supersymmetric pairs $$(\mathfrak {g},\mathfrak {k})$$ ( g , k ) , or more generally supersymmetric spaces G/K. We define certain invariant distributions on G/K, which we call ghost distributions, and which in some sense are induced from invariant distributions on $$G_0/K_0$$ G 0 / K 0 . Ghost distributions, and in particular their Harish-Chandra polynomials, give information about branching from G to a symmetric subgroup $$K'$$ K ′ which is related (and sometimes conjugate) to K. We discuss the case of $$G\times G/G$$ G × G / G for an arbitrary quasireductive supergroup G, where our results prove the existence of a polynomial which determines projectivity of irreducible G-modules. Finally, a generalization of Gorelik’s ghost centre is defined which we call the full ghost centre, $$\mathcal {Z}_{full}$$ Z full . For type I basic Lie superalgebras $$\mathfrak {g}$$ g we fully describe $$\mathcal {Z}_{full}$$ Z full , and prove that if $$\mathfrak {g}$$ g contains an internal grading operator, $$\mathcal {Z}_{full}$$ Z full consists exactly of those elements in $$\mathcal {U}\mathfrak {g}$$ U g acting by $$\mathbb {Z}$$ Z -graded constants on every finite-dimensional irreducible representation.

Factors Affecting Learners’ Decision to enrol in Commercial Subjects – A Case of the Chris Hani District, Eastern Cape Province, South Africa

Commercial subjects such as Accounting, Economics and Business Studies are an important stream in schools that help learners broaden their social and global knowledge. The purpose of the study was to identify the factors that influence learners’ decisions to enrol in commercial subjects. A case study served as the foundation for the study’s qualitative methodology. The study was based on Berger and Calabre’s (1983) Uncertainty Reduction Theory. The study purposely sampled ten learners from two secondary schools in the Chris Hani District, Eastern Cape Province, South Africa. Individual interviews were used to collect the data, which were then analysed using content analysis (CA) and categorised using thematic analysis. The findings proved that several factors, including learners’ ability in the subjects and the influence of teachers and parents, influenced their decision to enrol in commercial courses. The study recommends that learners should first be counselled, mentored, and assisted in making decisions about the choice of subjects. Lastly, schools should conduct aptitude and interest assessments. The contribution of this study is to provide insights on improving curriculum design and career guidance around commercial subjects in this region as well as to inform educational stakeholders on how to address potential barriers to enrolment in commercial subjects and lastly serve as a foundation for broader studies on commercial subject enrollment in South Africa. Keywords: Commercial Subjects; Learner’s decision; Subject Stream

Ternary Associativity and Ternary Lie Algebras at Cube Roots of Unity

We propose a new approach to extend the notion of commutator and Lie algebra to algebras with ternary multiplication laws. Our approach is based on the ternary associativity of the first and second kinds. We propose a ternary commutator, which is a linear combination of six triple products (all permutations of three elements). The coefficients of this linear combination are the cube roots of unity. We find an identity for the ternary commutator that holds due to the ternary associativity of either the first or second kind. The form of this identity is determined by the permutations of the general affine group GA(1,5)⊂S5. We consider this identity as a ternary analog of the Jacobi identity. Based on the results obtained, we introduce the concept of a ternary Lie algebra at cube roots of unity and provide examples of such algebras constructed using ternary multiplications of rectangular and three-dimensional matrices. We also highlight the connection between the structure constants of a ternary Lie algebra with three generators and an irreducible representation of the rotation group. The classification of two-dimensional ternary Lie algebras at cube roots of unity is proposed.