Representation Theory Research Articles

SPHERICAL REPRESENTATIONS FOR -FLOWS III: WEIGHT-EXTENDED BRANCHING GRAPHS

Abstract We apply Takesaki’s and Connes’s ideas on structure analysis for type III factors to the study of links (a short term of Markov kernels) appearing in asymptotic representation theory.

A generalized Selberg zeta function for flat space cosmologies

Flat space cosmologies (FSCs) are time dependent solutions of three-dimensional (3D) gravity with a vanishing cosmological constant. They can be constructed from a discrete quotient of empty 3D flat spacetime and are also called shifted-boost orbifolds. Using this quotient structure, we build a new and generalized Selberg zeta function for FSCs, and show that it is directly related to the scalar 1-loop partition function. We then propose an extension of this formalism applicable to more general quotient manifolds M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{M} $$\\end{document}/ℤ, based on representation theory of fields propagating on this background. Our prescription constitutes a novel and expedient method for calculating regularized 1-loop determinants, without resorting to the heat kernel. We compute quasinormal modes in the FSC using the zeroes of a Selberg zeta function, and match them to known results.

Quasi-hereditary covers of Temperley–Lieb algebras and relative dominant dimension

Many connections and dualities in representation theory and Lie theory can be explained using quasi-hereditary covers in the sense of Rouquier. Recent work by the first-named author shows that relative dominant (and codominant) dimensions are natural tools to classify and distinguish distinct quasi-hereditary covers of a finite-dimensional algebra. In this paper, we prove that the relative dominant dimension of a quasi-hereditary algebra, possessing a simple preserving duality, with respect to a direct summand of the characteristic tilting module is always an even number or infinite and that this homological invariant controls the quality of quasi-hereditary covers that possess a simple preserving duality. To resolve the Temperley–Lieb algebras, we apply this result to the class of Schur algebras $S(2, d)$ and their $q$ -analogues. Our second main result completely determines the relative dominant dimension of $S(2, d)$ with respect to $Q=V^{\otimes d}$ , the $d$ -th tensor power of the natural two-dimensional module. As a byproduct, we deduce that Ringel duals of $q$ -Schur algebras $S(2,d)$ give rise to quasi-hereditary covers of Temperley–Lieb algebras. Further, we obtain precisely when the Temperley–Lieb algebra is Morita equivalent to the Ringel dual of the $q$ -Schur algebra $S(2, d)$ and precisely how far these two algebras are from being Morita equivalent, when they are not. These results are compatible with the integral setup, and we use them to deduce that the Ringel dual of a $q$ -Schur algebra over the ring of Laurent polynomials over the integers together with some projective module is the best quasi-hereditary cover of the integral Temperley–Lieb algebra.

Symplectic Fourier–Deligne Transforms on G/U and the Algebra of Braids and Ties

Abstract We explicitly identify the algebra generated by symplectic Fourier–Deligne transforms (i.e., convolution with Kazhdan–Laumon sheaves) acting on the Grothendieck group of perverse sheaves on the basic affine space $G/U$, answering a question originally raised by A. Polishchuk. We show it is isomorphic to a distinguished subalgebra, studied by I. Marin, of the generalized algebra of braids and ties (defined in Type $A$ by F. Aicardi and J. Juyumaya and generalized to all types by Marin), providing a connection between geometric representation theory and an algebra defined in the context of knot theory. Our geometric interpretation of this algebra entails some algebraic consequences: we obtain a short and type-independent geometric proof of the braid relations for Juyumaya’s generators of the Yokonuma–Hecke algebra (previously proved case-by-case in types $A, D, E$ by Juyumaya and separately for types $B, C, F_{4}, G_{2}$ by Juyumaya and S. S. Kannan), a natural candidate for an analogue of a Kazhdan–Lusztig basis, and finally an explicit formula for the dimension of Marin’s algebra in Type $A_{n}$ (previously only known for $n \leq 4$).

Реконструкция медийных социальных представлений с использованием показателей статистики текста (на материале медиадискурса о пандемии)

<p style="text-align: justify;"><strong>Objective. </strong>The aim is to present an algorithm to reconstruct media social representations based on indicators of text statistics and to conduct a comparative analysis of the construction of semantically similar media joint ventures, such as “pandemic”, “coronavirus”, “COVID-19” in Russian media. <br><strong>Background</strong>. Social representations perform the most important functions in the process of social functioning of an individual and a group, serve as a tool for cognition, adaptation and regulation of behavior and are formed taking into account the influence of media social representations. Methods for studying social representations for various social groups are presented in psychological studies, however, methods for studying media social representations are discussed in single scientific work. The presented scientific project is based on the theory of social representations by S. Moskovici and generalizations by B. Hoyer regarding the construction of media social representations (naming, emotional attachment, thematic attachment, metaphorical attachment and attachment through basic antinomies). <br><strong>Study design</strong>. The phenomenon of the coronavirus pandemic, presented in media discourse, was used as the signified in the study. The signifier is a trio of semantically similar concepts (“pandemic”, “COVID”, “coronavirus”). <br><strong>Measurements</strong>. To reconstruct media social representations, statistically stable collocations were identified to indicate the measure of association, logically close to the associative experiment. Hence, it was possible to identify thematic networks, axiological and evaluative components, components-characteristics of activity. The research material is represented by texts about the COVID-19 pandemic (January 2020-March 2022: “Rossiyskaya Gazeta”: 19471 texts, 7,97 million words; “Kommersant”: 1482 texts, 1,07 million words, “Novaya Gazeta”: 705 texts, 0,9 million words) and processed using BootCat, TreeTagger, AntConc (lemmatization, frequency analysis). <br><strong>Results</strong>. The associative fields of the joint ventures are different and contain anchoring and objectification resources when using concepts, only few elements of the associative field are similar. Thus, depending on the concept used to signify the intent of the text, stories about the fear of infection, the treatment of the disease, or resistance to the harsh elements are created. <br><strong>Conclusions</strong>. The similar algorithm based on media text statistics can be used to reconstruct any media social representation.</p>

The politics of interaction and spect-actors in world rescue, eco, and plasticity: a Boalian perspective on digital environmental games

ABSTRACT In this paper, I explore the politics of engagement in environmental games from a political theatre theory perspective. Specifically, I focus on digital environmental games developed with classrooms in mind, and examine player interaction as politicized through the lens of Boalian theatre. To this end, I overview gaming as an intervention tool in the form of serious, environmental games. Next, I introduce the poetics of Boal’s TO, and connect its aim to restore theatre as a popular form of conversation and discussion to the aspiration of digital environmental games to develop an eco-friendly outlook. I build upon a rule-based approach to computer game rhetoric, and analyze how environmental games use rules as a rhetorical tool to create a pro-environmental statement. Lastly, I suggest that players in environmental games are not mere game players but rather ‘spect-actors’.

On the notion of the parabolic and the cuspidal support of smooth-automorphic forms and smooth-automorphic representations

AbstractIn this paper we describe several new aspects of the foundations of the representation theory of the space of smooth-automorphic forms (i.e., not necessarily $$K_\infty $$ K ∞ -finite automorphic forms) for general connected reductive groups over number fields. Our role model for this space of smooth-automorphic forms is a “smooth version” of the space of automorphic forms, whose internal structure was the topic of Franke’s famous paper (Ann Sci de l’ENS 2:181–279, 1998). We prove that the important decomposition along the parabolic support, and the even finer—and structurally more important—decomposition along the cuspidal support of automorphic forms transfer in a topologized version to the larger setting of smooth-automorphic forms. In this way, we establish smooth-automorphic versions of the main results of Franke and Schwermer (Math Ann 311:765–790, 1998) and of Mœglin and Waldspurger (Spectral Decomposition and Eisenstein Series, Cambridge University Press, 1995), III.2.6.

Au centre est l'État-nation. Le Monde vu par des étudiants du Kazakhstan

Abstract. A 2010 survey of more than 10 000 students from 18 countries showed that Central Asia was virtually absent from their representations of the world. In order to check whether this was also the case in the representations of students from this region, this survey was replicated in Kazakhstan, while nevertheless making the assumption that a Central Asian region would be largely represented by the students. While the results confirm important aspects of the theory of social representations of space, they also provide some original insights, showing in particular the very large place given to States, and in particular the Kazakh State, in the breakdown of the World. The prominence of States in Kazakh students' representations of the World can be analysed as a sign of the appropriation of the nation-state model by Kazakh students, some thirty years after the country's independence.

Kirillov Polynomials for the Exceptional Lie Algebra $$\mathfrak g_{2}$$

AbstractAs part of the development of the orbit method, Kirillov has counted the number of strictly upper triangular matrices with coefficients in a finite field of q elements and fixed Jordan type. One obtains polynomials with respect to q with many interesting properties and close relation to type A representation theory. In the present work, we develop the corresponding theory for the exceptional Lie algebra $$\mathfrak g_2$$ g 2 . In particular, we show that the leading coefficient can be expressed in terms of the Springer correspondence.

Algebra Environments II. Algebra Homomorphisms and Derivations

Algebra environments provide requisites for studying objects of interest in operator algebra theory, group representation theory, spin geometry, Clifford analysis, and several variable operator theory. The concept is analyzed by developing an algebraic geometry approach. Specific algebraic sets, called structure manifolds of algebra environments, and their Zariski tangent spaces are introduced and described by using as critical tools derivations on algebras. Structure manifolds of tensor environments in particular yield spaces of algebra homomorphisms. Consequently, such spaces could be investigated as algebraic manifolds. Related issues include characterizations of their Zariski tangent spaces and of derivations that preserve algebra homomorphisms.

The Basso-Dixon formula and Calabi-Yau geometry

We analyse the family of Calabi-Yau varieties attached to four-point fishnet integrals in two dimensions. We find that the Picard-Fuchs operators for fishnet integrals are exterior powers of the Picard-Fuchs operators for ladder integrals. This implies that the periods of the Calabi-Yau varieties for fishnet integrals can be written as determinants of periods for ladder integrals. The representation theory of the geometric monodromy group plays an important role in this context. We then show how the determinant form of the periods immediately leads to the well-known Basso-Dixon formula for four-point fishnet integrals in two dimensions. Notably, the relation to Calabi-Yau geometry implies that the volume is also expressible via a determinant formula of Basso-Dixon type. Finally, we show how the fishnet integrals can be written in terms of iterated integrals naturally attached to the Calabi-Yau varieties.

Exceptional sequences and rooted labeled forests

We give a representation-theoretic bijection between rooted labeled forests with [Formula: see text] vertices and complete exceptional sequences for the quiver of type [Formula: see text] with straight orientation. The ascending and descending vertices in the forest correspond to relatively injective and relatively projective objects in the exceptional sequence. We conclude that every object in an exceptional sequence for linearly oriented [Formula: see text] is either relatively projective or relatively injective or both. We construct a natural action of the extended braid group on rooted labeled forests and show that it agrees with the known action of the braid group on complete exceptional sequences. We also describe the action of [Formula: see text], the Garside element of the braid group, on rooted labeled forests using representation theory and show how this relates to cluster theory.

The Social Representation of Sustainable Mobility: An Exploratory Investigation on Social Media Networks

This paper analyzes the social representation (SR) of sustainable mobility as it emerges from social media networks. The role of social media (SM) as contexts of creation, negotiation, and diffusion of a mobility culture worldwide is discussed, together with the results of an empirical investigation that explored the contents and structure of the discourse on “sustainable mobility” as it emerges from the posts published (in English) on two highly used social media (Facebook and TikTok) platforms during the years 2022 and 2023. First of all, the results confirm the relevance of social media as an indicator of ongoing trends in the evolution of mobility culture and the usefulness of S. Moscovici’s theory of social representations as a theoretical framework for analyzing such trends. In particular, several new trends in the SRs of sustainable mobility were identified. These include, for example, the decline in skeptical views and the rise of more optimistic ones, regarding the feasibility of changing people’s mobility styles worldwide. Such views appear to be fostered by the positive perception of new technological innovations (electric vehicles), as well as by their endorsement by both the business sector and governmental institutions. Practical implications and theoretical indications for future research are also outlined.

Beliefs and practices of the nursing team related to pressure injury preventive measures: A analysis of social representations.

To analyse the process of elaborating social representations about pressure injury preventive measures by the nursing team (nurses and nurse technicians) and how this process relates to preventive practices for hospitalized patients. Qualitative study, with the application of the theory of social representations in its procedural methodological approach. The study was carried out in an inpatient clinic of a public hospital in the state of Rondônia, Brazil. Totally, 28 nursing professionals in the medical clinic sectors who had worked directly with patient care for more than 6 months participated. The data were collected between July and September 2021 via in-depth interviews with the application of a semi-structured instrument. Analysis was carried out with the help of ALCESTE software, which performed a lexicographic analysis, and also via thematic analysis. The COREQ guided the presentation of the research report. The social representations were developed based on the professionals' symbolic beliefs about the visibility/invisibility of the results of applying preventive care. These symbolic constructions mobilized positive and negative feelings among the nursing team, which guided the classification of prevention practices as being of greater or lesser priority among other care activities. There were favourable attitudes among professionals, which included applying prevention measures in their daily routines, and unfavourable attitudes of non-adherence to the institution's protocol for preventing pressure injuries. The nursing team's perception of pressure injury prevention is influenced by symbolic, affective, values, and social dimensions. Non-adherence behaviours are attributed to the belief in the invisibility of prevention outcomes, resulting in a reluctance to implement preventive measures. Understanding the subjective logic that explains the thinking and actions of the nursing team suggests the need to incorporate discussions on beliefs, values, sentiments, and attitudes of nursing professionals into educational programs on pressure injury prevention. No public contribution.

Prestack sparse envelope seismic inversion method adopting the L0−L2-norm regularization

The high drilling costs limit the possibility of obtaining reliable low-frequency constrained models in marine environments or areas with sparse well-log data, making it difficult to constrain lateral continuity and deep data. To enhance stability and accuracy in prestack seismic inversion in these areas, an elastic parameter estimation approach using sparse envelope inversion with [Formula: see text]-norm regularization is proposed. The method combines signal sparse representation and modulation theories to derive a new formula for sparse envelope extraction at lower frequencies. By applying [Formula: see text]-norm regularization to the sparse envelope, we obtain parameter inversion results with smoothed trends, augmenting low-wavenumber information for improved model constraints. In addition, taking the envelope inversion results obtained by the [Formula: see text] norm as the model constraint, the sparsest inversion results with obvious block-like characteristics are obtained by regularizing the inversion equation with the [Formula: see text] norm. Notably, the new method effectively suppresses wavelet side lobes, resulting in stable and accurate inversions without the need for initial low-frequency models. A synthetic example is used to illustrate the feasibility and stability of proposed approach and further demonstrate its practicality in reservoir parameter estimation through a field case study of gas exploration.

Generation of Visual Representations for Multi-Modal Mathematical Knowledge

In this paper we introduce MaRE, a tool designed to generate representations in multiple modalities for a given mathematical problem while ensuring the correctness and interpretability of the transformations between different representations. The theoretical foundation for this tool is Representational Systems Theory (RST), a mathematical framework for studying the structure and transformations of representations. In MaRE’s web front-end user interface, a set of probability equations in Bayesian Notation can be rigorously transformed into Area Diagrams, Contingency Tables, and Probability Trees with just one click, utilising a back-end engine based on RST. A table of cognitive costs, based on the cognitive Representational Interpretive Structure Theory (RIST), that a representation places on a particular profile of user is produced at the same time. MaRE is general and domain independent, applicable to other representations encoded in RST. It may enhance mathematical education and research, facilitating multi-modal knowledge representation and discovery.

Coloring History

Since its release, the Netflix show, Bridgerton, has been a hit series among its mass audiences as it harnesses an average of 103,550,000 views per season and currently holds fourth place in Netflix’s most popular English-language series (Tudum, 2024). The show has gained immense popularity not only because of its captivating plotlines but also because of the diverse cast of characters placed within the Regency Era. From this, it is critical to dissect and understand how race is represented within the show to discern the discourses being perpetuated to its viewers. Throughout this essay, I will employ Stuart Hall’s theory of representation and the politics of signification to unpack the racial representation within the Netflix show, Bridgerton, and how it reproduces colour-blind casting, interracial relationships, and dark versus light symbolism to perpetuate post-racial and Orientalist discourses. This analysis explores how colour-blind casting sustains post-racial discourses in the disguise of escapist media, which strips the Other of their lived experience. Moreover, it highlights how the depiction of interracial relationships within the show contributes to both post-racial and Orientalist discourses by placing the Occident over the Orient, and disregarding the prejudice often faced by interracial couples during that period. It further reveals how dark versus light symbolism is embedded throughout the interactions between multiple characters, which elicits the disruption of the white imaginary by Black individuals. As the show progresses and continues to release seasons that reach such large audiences, it is crucial to analyze what discourses surrounding race are being perpetuated and its potential impact on reinforcing stereotypes and misrepresenting marginalized communities.

Randomness, Integrability and Universality

In spring 2022, the Galileo Galilei Institute for Theoretical Physics hosted a seven-week Workshop on ‘Randomness, Integrability and Universality’. The Workshop addressed a series of questions on exactly solvable models of statistical mechanics, having numerous ties and overlaps with various problems in random matrix models, probability, representation theory, and combinatorics. Much recent progress in these areas exploits the underlying notion of quantum integrability. Here we report on the scientific motivations and background of this activity and on its main outputs.

COVID-19 and emerging adults in India: A social representation approach

The outbreak and rapid spread of the novel coronavirus disease or COVID-19 had health, social and economic ramifications for people worldwide. Using the framework of Social Representation Theory, the present research attempts to understand how emerging adults (18–29 years) made sense of the COVID-19 outbreak, the role of collectives (organizations) in managing it, and the evolution of this understanding over time. The function of this theoretical perspective in the present study is to help make sense of the COVID-19 pandemic as a threatening event and in the process understand if responsibility or blame is attributed to specific collectives. Data was collected in two phases, from ten participants (four males and six females), aged 18–29 years residing in India. Semi-structured interviews were used to generate data. Findings have been discussed as ‘sociogenesis of COVID-19’, ‘role of collectives: heroes, villains, or on the fence’, ‘collective symbolic coping: making sense of collective threats’, and ‘the next normal: learnings from COVID-19). Finally, this study would have implications for research on social representation and its evolution, especially of emerging infectious diseases and the representation of collectives may also be used to reevaluate health policies and get better prepared for future emergencies.

Responding to misrepresentation: sample pack creation, branding and distribution

ABSTRACT This article documents the creation and distribution of a new sample pack, Instruments INDIA, as an intervention against systemic orientalist practices observable within the marketing, branding and production of non-Western instrument sample packs. This intervention draws upon Edward Said’s model of Orientalism and Stuart Hall’s theories of representation, fortifying an interrogation of practices that misrepresent diverse sonic content, musicians and cultures in negative ways. The research brings into focus the influence digital samplers and early sample library CDs had in determining technical configurations and misrepresentation tendencies found within paratextual elements of contemporary sample packs. The article joins forces with a leading sample pack distributor, Loopmasters, to cultivate change via new policy documents and good practice principles. Attending to problematic representation issues at the start of the music production chain offers a strategy to scaffold and promote positive change throughout the production process and into other connected music industry areas, where influential language and imagery align sample content with music production goals and aspirations. The article offers a first-hand account of sample pack development in collaboration with musicians of Indian musical instruments, which provide a case study to better understand and inform more ethical commercial practices in the future.