Irreducible Representations Research Articles

Robust Sparse Sufficient Dimension Reduction via Adaptive Lasso Penalty

In some multi-index models applications, there is an important role for dimension reduction and variable selection (VS) methods. The ALMAVE is a method for variables selection under sufficient dimension reduction theory settings. It combines the adaptive Lasso and MAVE (minimum average variance estimation) to produce sparse and accurate solutions. The ALMAVE is a very sensitive method to outliers in the response y due to use the least-squares criterion. In this article, we proposed robust ALMAVE. Also, an efficient estimation algorithm was proposed. The simulation studies and Logo design data analysis were employed to check the effectiveness of ALMAVE.

Branching patterns of wave trains in mass-in-mass lattices

We investigate the existence and branching patterns of wave trains in the mass-in-mass (MiM) lattice, which is a variant of the Fermi–Pasta–Ulam (FPU) lattice. In contrast to FPU lattice, we have to solve coupled advance-delay differential equations, which are reduced to a finite-dimensional bifurcation equation with an inherited Hamiltonian structure by applying a Lyapunov–Schmidt reduction and invariant theory. We establish a link between the MiM lattice and the monatomic FPU lattice. That is, the monochromatic and bichromatic wave trains persist near $\mu =0$ in the nonresonance case and in the resonance case $p:q$ where $q$ is not an integer multiple of $p$ . Furthermore, we obtain the multiplicity of bichromatic wave trains in $p:q$ resonance where $q$ is an integer multiple of $p$ , based on the singular theorem.

Some Results of Representation of the Character Table of S_6

n this essay we use standard Young tableaux to compute each equivalent irreducible representation of the Symmetric group by determining the matrices for adjacent transpositions (12) , (23) , (34), (45) and (56) of .

L-packets over strong real forms

Langlands [On the classification of irreducible representations of real algebraic groups, Math. Surveys Monogr., vol. 31, Amer. Math. Soc., Providence, RI, 1989, pp. 101–170] defined L L -packets for real reductive groups. In order to refine the local Langlands correspondence, Adams-Barbasch-Vogan [The Langlands classification and irreducible characters for real reductive groups, Progress in Mathematics, vol. 104, Birkhäuser Boston, Inc., Boston, MA, 1992] combined L-packets over all real forms belonging to an inner class. In the tempered setting, using different methods, Kaletha [Ann. of Math. (2) 184 (2016), pp. 559–632] also defines such combined L-packets with a refinement to the local Langlands correspondence. We prove that the tempered L-packets of Adams-Barbasch-Vogan and Kaletha are the same and are parameterized identically.

Manifestly covariant worldline actions from coadjoint orbits. Part I. Generalities and vectorial descriptions

We derive manifestly covariant actions of spinning particles starting from coadjoint orbits of isometry groups, by using Hamiltonian reductions. We show that the defining conditions of a classical Lie group can be treated as Hamiltonian constraints which generate the coadjoint orbits of another, dual, Lie group. In case of (inhomogeneous) orthogonal groups, the dual groups are (centrally-extended inhomogeneous) symplectic groups. This defines a symplectic dual pair correspondence between the coadjoint orbits of the isometry group and those of the dual Lie group, whose quantum version is the reductive dual pair correspondence à la Howe. We show explicitly how various particle species arise from the classification of coadjoint orbits of Poincaré and (A)dS symmetry. In the Poincaré case, we recover the data of the Wigner classification, which includes continuous spin particles, (spinning) tachyons and null particles with vanishing momenta, besides the usual massive and massless spinning particles. In (A)dS case, our classification results are not only consistent with the pattern of the corresponding unitary irreducible representations observed in the literature, but also contain novel information. In dS, we find the presence of partially massless spinning particles, but continuous spin particles, spinning tachyons and null particles are absent. The AdS case shows the largest diversity of particle species. It has all particles species of Poincaré symmetry except for the null particle, but allows in addition various exotic entities such as one parameter extension of continuous particles and conformal particles living on the boundary of AdS. Notably, we also find a large class of particles living in “bitemporal” AdS space, including ones where mass and spin play an interchanged role. We also discuss the relative inclusion structure of the corresponding orbits.

On the temperature dependence of transition intensities of rare-earth ions: A modified Judd-Ofelt theory

We report on a modification of the Judd-Ofelt theory accounting for thermal populations of Stark sub-levels of the multiplets involved in transitions in absorption and emission. The calculations require the data on the crystal-field splitting, the irreducible representations of Stark sub-levels and the selection rules for electric-dipole transitions. Simple formulas for absorption oscillator strengths and probabilities of radiative spontaneous transitions are obtained. The application of the proposed model allowed us to remove the contradiction between the transition intensities in absorption and emission (3H6 ↔ 3F4) for Tm3+-doped crystals. It was also used to explain the experimental temperature dependences of luminescence lifetimes of the 3F4 Tm3+ manifold in two widely used laser crystals, LiYF4 and Y3Al5O12. The case of Yb3+ ions featuring a small number of crystal-field sub-levels is also considered.

Complexity theory for complexity reduction? Revisiting the ontological and epistemological basis of complexity science with Critical Realism

AbstractComplexity theory (CT) identifies our social system as a contingent and emergent product of non‐linear interactions between existing patterns and events. However, CT scholars carrying out various empirical applications have often adopted constructivist positions that disallow the separate existence of social systems and agency, thereby preventing effective analysis of their interactions. Instead, with the help of Critical Realism (CR), we offer a realist complexity approach that sees complexity in terms of the distinction between the domains of the Real, the Actual, and the Empirical, when existing studies of CT still work with a flat ontology that collapses the three domains into one (the Empirical domain). Our non‐conflationary CR‐CT approach thus argues that a satisfactory explanation of social complexity cannot be at the level of agential experience (the Empirical domain) or at the level of human and systematic events (the Actual domain) but needs to identify causal mechanisms (in the Real domain) of such events. It then combines this depth ontology (that distinguishes the three reality domains) with epistemological relativism (that underscores the contingent character of knowledge claims) to argue that though our knowledge and complexity reduction techniques are socially constructed, it hardly follows that the ontological dimension of reality (spreading across the three domains) is always affected by our complexity reduction efforts at the epistemological dimension in the Empirical domain.

Notes on gauge fields and discrete series representations in de Sitter spacetimes

In this note we discuss features of the simplest spinning Discrete Series Unitary Irreducible Representations (UIR) of SO(1,4). These representations are known to be realised in the single particle Hilbert space of a free gauge field propagating in a four dimensional fixed de Sitter background. They showcase distinct features as compared to the more common Principal Series realised by heavy fields. Upon computing the 1 loop Sphere path integral we show that the edge modes of the theory can be understood in terms of a Discrete Series of SO(1, 2). We then canonically quantise the theory and show how group theory constrains the mode decomposition. We further clarify the role played by the second SO(4) Casimir in the single particle Hilbert space of the theory.

Bounds on multiplicities of symmetric pairs of finite groups

Abstract Let $\Gamma $ be a finite group, let $\theta $ be an involution of $\Gamma $ and let $\rho $ be an irreducible complex representation of $\Gamma $ . We bound ${\operatorname {dim}} \rho ^{\Gamma ^{\theta }}$ in terms of the smallest dimension of a faithful $\mathbb {F}_p$ -representation of $\Gamma /\operatorname {\mathrm {Rad}}_p(\Gamma )$ , where p is any odd prime and $\operatorname {\mathrm {Rad}}_p(\Gamma )$ is the maximal normal p-subgroup of $\Gamma $ . This implies, in particular, that if $\mathbf {G}$ is a group scheme over $\mathbb {Z}$ and $\theta $ is an involution of $\mathbf {G}$ , then the multiplicity of any irreducible representation in $C^\infty \left( \mathbf {G}(\mathbb {Z}_p)/ \mathbf {G} ^{\theta }(\mathbb {Z}_p) \right)$ is bounded, uniformly in p.

Smith theory and cyclic base change functoriality

Abstract Lafforgue and Genestier-Lafforgue have constructed the global and (semisimplified) local Langlands correspondences for arbitrary reductive groups over function fields. We establish various properties of these correspondences regarding functoriality for cyclic base change: For $\mathbf {Z}/p\mathbf {Z}$ -extensions of global function fields, we prove the existence of base change for mod p automorphic forms on arbitrary reductive groups. For $\mathbf {Z}/p\mathbf {Z}$ -extensions of local function fields, we construct a base change homomorphism for the mod p Bernstein center of any reductive group. We then use this to prove existence of local base change for mod p irreducible representation along $\mathbf {Z}/p\mathbf {Z}$ -extensions, and that Tate cohomology realizes base change descent, verifying a function field version of a conjecture of Treumann-Venkatesh. The proofs are based on equivariant localization arguments for the moduli spaces of shtukas. They also draw upon new tools from modular representation theory, including parity sheaves and Smith-Treumann theory. In particular, we use these to establish a categorification of the base change homomorphism for mod p spherical Hecke algebras, in a joint appendix with Gus Lonergan.

On a vector boson model realization of the proxy-SU(3) symmetry

We test the capability of highest-weight (h.w.) irreducible representations (irreps) (λ, µ)hw determined in the model scheme with proxy SU(3) symmetry to describe groundand γ- collective bands and transition rates in heavy even-even nuclei when applied within the vector boson model (VBM) with broken SU(3) symmetry. We find that for nuclei with closely disposed groundand γ-band energy levels the obtained root-mean square (RMS) deviations between theory and experiment exceed those obtained in the original VBM application in which the (λ, µ) free values are freely adjusted to data. In nuclei with a large ground-γ energy distance we find that the quality of the (λ, µ)hw description coincides with that obtained by (λ, µ) free. This preliminary result suggests that the proxy SU(3) symmetry may be realized in terms of the algebraic VBM.

Orthogonal magnetic structures of Fe4O5: representation analysis and DFT calculations.

The magnetic and electronic structures of Fe4O5 have been investigated at ambient and high pressures via a combination of representation analysis, density functional theory (DFT+U) calculations, and Mössbauer spectroscopy. A few spin configurations corresponding to the different irreducible representations have been considered. The total-energy calculations reveal that the magnetic ground state of Fe4O5 corresponds to an orthogonal spin order. Depending on the magnetic propagation vector k, two spin-ordered phases with minimal energy differences are realized. The lowest energy magnetic phase is related to k = (0, 0, 0) and is characterized by ferromagnetic ordering of iron magnetic moments at prismatic sites along the b-axis and antiferromagnetic ordering of iron moments at octahedral sites along the c-axis. For the k = (1/2, 0, 0) phase, the moments in the prisms are antiferromagnetically ordered along the b-axis and the moments in the octahedra are still antiferromagnetically ordered along the c-axis. Under high pressure, Fe4O5 exhibits magnetic transitions with the corresponding electronic transitions of the metal-insulator type. At a critical pressure PC ∼ 60 GPa, the Fe ions at the octahedral sites undergo a high-spin to low-spin state crossover with a decrease in the unit-cell volume of ∼4%, while the Fe ions at the prismatic sites remain in the high-spin state up to 130 GPa. This site-dependent magnetic collapse is experimentally observed in the transformation of Mössbauer spectra measured at room temperature and high pressures.

Models of representations for classical series of Lie algebras

By a model of representations of a Lie algebra we mean a representation which is a direct sum of all irreducible finite-dimensional representations taken with multiplicity $1$. An explicit construction of a model of representations for all classical series of simple Lie algebras is given. This construction is generic for all classical series of Lie algebras. The space of the model is constructed as the space of polynomial solutions of a system of partial differential equations, where the equations are constructed form relations between minors of matrices taken from the corresponding Lie group. This system admits a simplification very close to the GKZ system, which is satisfied by $A$-hypergeometric functions.

Reduction of Sufficient Conditions in Variational Obstacle Avoidance Problems

This paper studies Sufficient conditions in a variational obstacle avoidance problem on complete Riemannian manifolds. That is, we minimize an action functional, among a set of admissible curves, which depends on an artificial potential function used to avoid obstacles. We provide necessary and Sufficient conditions under which the resulting critical points—the so-called modified Riemannian cubics—are local minimizers. We then study the theory of reduction by symmetries of Sufficient conditions for optimality in variational obstacle avoidance problems on Lie groups endowed with a left-invariant metric. This amounts to left-translating the Bi-Jacobi fields described to the Lie algebra, and studying the corresponding bi-conjugate points. New conditions are provided in terms of the invertibility of a certain matrix.

Local parameters of supercuspidal representations

Abstract For a connected reductive group G over a nonarchimedean local field F of positive characteristic, Genestier-Lafforgue and Fargues-Scholze have attached a semisimple parameter ${\mathcal {L}}^{ss}(\pi )$ to each irreducible representation $\pi $ . Our first result shows that the Genestier-Lafforgue parameter of a tempered $\pi $ can be uniquely refined to a tempered L-parameter ${\mathcal {L}}(\pi )$ , thus giving the unique local Langlands correspondence which is compatible with the Genestier-Lafforgue construction. Our second result establishes ramification properties of ${\mathcal {L}}^{ss}(\pi )$ for unramified G and supercuspidal $\pi $ constructed by induction from an open compact (modulo center) subgroup. If ${\mathcal {L}}^{ss}(\pi )$ is pure in an appropriate sense, we show that ${\mathcal {L}}^{ss}(\pi )$ is ramified (unless G is a torus). If the inducing subgroup is sufficiently small in a precise sense, we show $\mathcal {L}^{ss}(\pi )$ is wildly ramified. The proofs are via global arguments, involving the construction of Poincaré series with strict control on ramification when the base curve is ${\mathbb {P}}^1$ and a simple application of Deligne’s Weil II.

On irreducible representations of the free group

On irreducible representations of the free group

$ \mathcal{N} = 2 $ double graded supersymmetric quantum mechanics via dimensional reduction

<abstract><p>We presented a novel $ \mathcal{N} = 2 $ $ \mathbb{Z}_2^2 $-graded supersymmetric quantum mechanics ($ {\mathbb{Z}_2^2} $-SQM) which has different features from those introduced so far. It is a two-dimensional (two-particle) system and was the first example of the quantum mechanical realization of an eight-dimensional irreducible representation (irrep) of the $ \mathcal{N} = 2 $ $ \mathbb{Z}_2^2 $-supersymmetry algebra. The $ {\mathbb{Z}_2^2} $-SQM was obtained by quantizing the one-dimensional classical system derived by dimensional reduction from the two-dimensional $ {\mathbb{Z}_2^2} $-supersymmetric Lagrangian of $ \mathcal{N} = 1 $, which we constructed in our previous work. The ground states of the $ {\mathbb{Z}_2^2} $-SQM were also investigated.</p></abstract>

Benefits for emotional regulation of contact with nature: a systematic review.

Exposure to natural environments, such as parks, forests, and green areas, is often linked to a decrease in stress, anxiety and depression, while contributing to improved emotional wellbeing. These observations are supported by well-established theories, such as the Stress Reduction Theory and Attention Restoration Theory, which highlight the psychological benefits of interacting with nature. However, the relationship between exposure to nature and emotions, and in particular, with emotional regulation, is an evolving aspect of research with no clear conclusions. Emotional regulation can be deliberate in nature, where individuals voluntarily participate in modifying various aspects of their emotions, such as their type, intensity, quality or duration. Alternatively, it may be automatic, originating from sensory perception and acting without full awareness, but significantly influencing emotional experiences. In this context, the environmental self-regulation hypothesis, suggests that people consciously or unconsciously use their physical environment to regulate their emotions. To analyze the evidence of the relationship between contact with nature and emotional regulation, we conducted this review. Using the PRISMA statement as a reference, we conducted keyword searches in five databases in the period between 2013 and 2023. The databases selected were Scopus, Web of Science (WoS), PubMed, PsycINFO and ScienceDirect. In addition, a manual search was carried out of journals in the research field. Initially, from which gray literature, reviews and duplicates were removed in a first step. The resulting articles were then filtered using their titles and abstracts. Subsequently, the abstracts of the 25 selected articles were reviewed and discussed by researchers to reach a final decision based on consensus about the adequacy of each paper. Finally, nine articles were included in the systematic review. In general terms, this review suggests that research on the relationship between contact with nature and emotional regulation provides valuable insights into how natural environments can contribute to the emotional wellbeing and physical and mental health of the population.

“Not Liking the Look of This”: Examining the Relationship Between COVID-Related Social Media Exposure and Mental Health Symptoms

“Not Liking the Look of This”: Examining the Relationship Between COVID-Related Social Media Exposure and Mental Health Symptoms

Examining the Impact of Emojis on Disaster Communication: A Perspective from the Uncertainty Reduction Theory

Examining the Impact of Emojis on Disaster Communication: A Perspective from the Uncertainty Reduction Theory