Superconformal field theories (SCFTs) are an important class of quantum field theories. These SCFTs have been a significant component in exploring and comprehending the fundamental framework of quantum field theories. In the vast realm of quantum field theories, integrability plays a crucial role, providing powerful analytic tools that allow us to solve certain physical quantities exactly. In this thesis, we focus on the representation theory of the algebraic structure in six-dimensional (6D) SCFTs and investigate the intricate interplay between 6D SCFTs and integrability. To begin, we delve into the fundamental concepts of representation theory, establishing a solid foundation for our subsequent analysis. The discussion then will move on to all possible generators in the SCFTs, explaining how they are realized in terms of bosonic and fermionic oscillators. Finally, we investigate spin chains and their application in 6D SCFTs. We demonstrate that symmetry arguments derived from representation theory are not sufficient to establish the integrability of the spin chains in 6D SCFTs. This conclusion does not imply the absence of integrable systems within 6D SCFTs; rather, it suggest there are other potential methods available e.g., correlation functions, to explore the appearance of integrable systems in 6D SCFTs. / Superkonforma fältteorier (SCFTs) är en viktig klass av kvantfältteorier. Dessa SCFTs utgör en viktig komponent för att utforska och förstå det fundamentala ramverket för kvantfältteorin. Inom det stora riket av kvantfältteori spelar integrabilitet en avgörande roll, vilket tillhandahåller kraftfulla analytiska verktyg som gör att vi kan lösa vissa fysiska storheter exakt. I denna avhandling fokuserar vi på representationsteorin av den algebraiska strukturen i sexdimensionella (6D) SCFTs och undersöker det intrikat samspelet mellan 6D SCFTs och integrabilitet. Till att börja med kommer vi att fördjupa oss i de grundläggande begreppen inom representationsteori och skapa en gedigen grund för vår efterföljande analys.  Diskussionen kommer sedan att gå vidare till alla möjliga generatorer i SCFTs, och förklarar hur de realiseras i termer av bosoniska och fermioniska oscillatorer. Slutligen kommer spinnkedjor och dess tillämpningar i 6D SCFTs att undersökas. Vi kommer visa att symmetriargument som härleds från representationsteori inte är  tillräckliga för att fastställa integrerbarhet av spinnkedjor i 6D SCFTs. Denna slutsats innebär inte att integrerbara system inte existerar inom 6D SCFTs, utan föreslår att det finns andra potentiella metoder, till exempel korrelationsfunktioner, för att utforska existensen av integrerbara system i 6D SCFTs.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第23685号 / 理博第4775号 / 新制||理||1684(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 荒川 知幸, 教授 玉川 安騎男, 教授 並河 良典 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Using the representation theory of positive definite sequences some propositions in additive number theory are obtained and H. Bohr's approximation theorem is deduced. A unified approach to theorems by S. Bochner, S, N, Bernstein and H. Hamburger is discussed and some operator versions of numerical moment problems are studied. The Appendix contains comments to J. von Neumann's spectral theorem for self-adjoint operators in Hilbert space. / Thesis / Doctor of Philosophy (PhD)

We provide a combinatorial interpretation of the frequency of any irreducible representation of Sn in representations of Sn arising from group actions on words. Recognizing that representations arising from group actions naturally split across orbits yields combinatorial interpretations of the irreducible decompositions of representations from similar group actions. The generalization from group actions on words to group actions on matrices gives rise to representations that prove to be much less transparent. We share the progress made thus far on the open problem of determining the irreducible decomposition of certain representations of Sm × Sn arising from group actions on matrices.

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“Can one hear the shape of a drum?” This seemingly innocent question spawned a lot of research in the early 20th century. Even though the answer is “No, we can't”, we can hear the volume. This is known as Weyl's Law. In a more modern context, we can use new methods to study similar questions. More precisely, we can study locally symmetric spaces and the algebra of invariant differential operators. Generalizing the above, we can incorporate Hecke operators and find asymptotic formulas for their traces. We study this problem in a global context, namely if the underlying group is the group of adelic points of a quasi-split, simple reductive group. Our main tool is the Arthur-Selberg trace formula. The spectral side is dealt with, utilizing a condition on the normalizing factors of certain intertwining operators. The geometric side is more complicated and needs a more refined analysis. Most importantly, the test functions need to be specifically crafted to ensure compact support on the one hand, and sufficiently strong estimates on the other. The resulting geometric side can be split according to the Bruhat decomposition and treated separately, using various methods from reduction theory to algebraic and analytic number theory.

This thesis explores basic representation theory of finite groups, covering basic definitions such as irreducible representations. The main part of the work focuses on finding irreducible representations of automorphism groups of simple graphs, in particular for graphs consisting of two identical odd cycle components by using the knowledge of the automorphism group of cycle graphs. Character theory is used to find the irreducible representations.

No / This paper is concerned with a variety of contemporary representations of dementia in both mainstream made-for-box-office cinema and in TV soap and drama. Such representations frequently draw on familiar tropes of global memory loss, violence and aggression, extreme dependency on heroic carers, catastrophic prognosis, and early death. Whilst such narrative devices may be excusable to some extent in film made for purposes of entertainment, the producers have considerable responsibility for public awareness and understandings of dementia, which, we would argue, should be discharged in a socially responsible way, rather than purely in order to achieve dramatic effect. Moreover, it has been widely argued in recent years (eg Alexander et al 2005) that film of this nature can be used ¿as it stands¿ in the education of health and social care practitioners. Instead, we would argue that students and practitioners need to learn the basic principles of representation theory, in order to understand and critique how film works to influence and socially construct views of health, illness and regimes of truth around them, paying central attention always to the question of whose interests are served by the representation in question.

This thesis concerns the use of perverse sheaves with coefficients in commutative rings and in particular, fields of positive characteristic, in the study of modular representation theory. We begin by giving a new geometric interpretation of classical connections between the representation theory of the general linear groups and symmetric groups. We then survey work, joint with D. Juteau and G. Williamson, in which we construct a class of objects, called parity sheaves. These objects share many properties with the intersection cohomology complexes in characteristic zero, including a decomposition theorem and a close relation to representation theory. The final part of this document consists of two computations of IC stalks in the nilpotent cones of sl₃and sl₄. These computations build upon our calculations in sections 3.5 and 3.6 of (31), but utilize slightly more sophisticated techniques and allow us to compute the stalks in the remaining characteristics. / text