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Zoos as an institution have a significant international and educational reach, with hundreds of institutions worldwide and hundreds of millions of visitors each year. But even with such reach, zoos continue to be overlooked by the field of literature. Over the years, zoos have been represented in many literary landscapes, and the same can be analyzed for understanding how zoos as an institution. Furthermore, the zoo’s representation in literature opens the venue for discourses on ethical considerations and moral dilemmas surrounding zoos, and questioning the impact of human intervention on the natural world. Authors of such work navigate themes of conservation, animal rights, and the blurred lines between entertainment and exploitation, prompting readers to reconsider their perceptions of zoological spaces. This research paper delves into the multifaceted representations of zoos in literature, examining how authors have engaged with the concept of the zoo as a cultural and societal institution. By employing a thematic approach, the paper investigates how zoos have been depicted as microcosms of society, mirrors reflecting human behavior and societal attitudes towards nature. The exploration encompasses works, such as Yann Martel's Life of Pi and Edward Albee’s The Zoo Story which delve into the dichotomy between captivity and freedom. The study spans various literary works aiming to provide a comprehensive analysis of the evolving portrayal of zoos in literature. [ABSTRACT FROM AUTHOR]

This paper performed an investigation into the s-embedding of the Lie superalgebra (→ S 1 ∣ 1 ) , a representation of smooth vector fields on a (1,1)-dimensional super-circle. Our primary objective was to establish a precise definition of the s-embedding, effectively dissecting the Lie superalgebra into the superalgebra of super-pseudodifferential operators ( S ψ D ⊙ ) residing on the super-circle S 1 | 1 . We also introduce and rigorously define the central charge within the framework of (→ S 1 ∣ 1 ) , leveraging the canonical central extension of S ψ D ⊙ . Moreover, we expanded the scope of our inquiry to encompass the domain of fuzzy Lie algebras, seeking to elucidate potential connections and parallels between these ostensibly distinct mathematical constructs. Our exploration spanned various facets, including non-commutative structures, representation theory, central extensions, and central charges, as we aimed to bridge the gap between Lie superalgebras and fuzzy Lie algebras. To summarize, this paper is a pioneering work with two pivotal contributions. Initially, a meticulous definition of the s-embedding of the Lie superalgebra (→ S 1 | 1 ) is provided, emphasizing the representationof smooth vector fields on the (1,1)-dimensional super-circle, thereby enriching a fundamental comprehension of the topic. Moreover, an investigation of the realm of fuzzy Lie algebras was undertaken, probing associations with conventional Lie superalgebras. Capitalizing on these discoveries, we expound upon the nexus between central extensions and provide a novel deformed representation of the central charge. [ABSTRACT FROM AUTHOR]

Important connections in representation theory arise from resolving a finite-dimensional algebra by an endomorphism algebra of a generator-cogenerator with finite global dimension; for instance, Auslander's correspondence, classical Schur–Weyl duality and Soergel's Struktursatz. Here, the module category of the resolution and the module category of the algebra being resolved are linked via an exact functor known as the Schur functor. In this paper, we investigate how to measure the quality of the connection between module categories of (projective) Noetherian algebras, B, and module categories of endomorphism algebras of generator-relative cogenerators over B which are split quasi-hereditary Noetherian algebras. In particular, we are interested in finding, if it exists, the highest degree n so that the endomorphism algebra of a generator-cogenerator provides an n-faithful cover, in the sense of Rouquier, of B. The degree n is known as the Hemmer–Nakano dimension of the standard modules. We prove that, in general, the Hemmer–Nakano dimension of standard modules with respect to a Schur functor from a split highest weight category over a field to the module category of a finite-dimensional algebra B is bounded above by the number of non-isomorphic simple modules of B. We establish methods for reducing computations of Hemmer–Nakano dimensions in the integral setup to computations of Hemmer–Nakano dimensions over finite-dimensional algebras, and vice-versa. In addition, we extend the framework to study Hemmer–Nakano dimensions of arbitrary resolving subcategories. In this setup, we find that the relative dominant dimension over (projective) Noetherian algebras is an important tool in the computation of these degrees, extending the previous work of Fang and Koenig. In particular, this theory allows us to derive results for Schur algebras and the BGG category \mathcal {O} in the integral setup from the finite-dimensional case. More precisely, we use the relative dominant dimension of Schur algebras to completely determine the Hemmer–Nakano dimension of standard modules with respect to Schur functors between module categories of Schur algebras over regular Noetherian rings and module categories of group algebras of symmetric groups over regular Noetherian rings. We exhibit several structural properties of deformations of the blocks of the Bernstein-Gelfand-Gelfand category \mathcal {O} establishing an integral version of Soergel's Struktursatz. We show that deformations of the combinatorial Soergel's functor have better homological properties than the classical one. [ABSTRACT FROM AUTHOR]

Young diagrams are fundamental combinatorial objects in representation theory and algebraic geometry. Many constructions that rely on these objects depend on variations of a straightening process that expresses a filling of a Young diagram as a sum of semistandard tableaux subject to certain relations. This paper solves the long standing open problem of giving a non-iterative formula for straightening a filling. We apply our formula to give a complete generalization of a theorem of Gonciulea and Lakshmibai. [ABSTRACT FROM AUTHOR]

Let D n , k be the set of all permutations of the symmetric group S n that have no cycles of length i for all 1 ≤ i ≤ k . In the paper mentioned above, Ku, Lau, and Wong prove that the set of all the largest independent sets of the Cayley graph Cay (S n , D n , k) is equal to the set of all the largest independent sets in the derangement graph Cay (S n , D n , 1) , provided n is sufficiently large in terms of k. We give a simpler proof that holds for all n, k and also applies to the alternating group. [ABSTRACT FROM AUTHOR]

Davydov–Yetter (DY) cohomology classifies infinitesimal deformations of the monoidal structure of tensor functors and tensor categories. In this paper we provide new tools for the computation of the DY cohomology for finite tensor categories and exact functors between them. The key point is to realize DY cohomology as relative Ext groups. In particular, we prove that the infinitesimal deformations of a tensor category C are classified by the 3-rd self-extension group of the tensor unit of the Drinfeld center Z (C) relative to C . From classical results on relative homological algebra we get a long exact sequence for DY cohomology and a Yoneda product for which we provide an explicit formula. Using the long exact sequence and duality, we obtain a dimension formula for the cohomology groups based solely on relatively projective covers which reduces a problem in homological algebra to a problem in representation theory, e.g. calculating the space of invariants in a certain object of Z (C) . Thanks to the Yoneda product, we also develop a method for computing DY cocycles explicitly which are needed for applications in the deformation theory. We apply these tools to the category of finite-dimensional modules over a finite-dimensional Hopf algebra. We study in detail the examples of the bosonization of exterior algebras Λ C k ⋊ C [ Z 2 ] , the Taft algebras and the small quantum group of sl 2 at a root of unity. [ABSTRACT FROM AUTHOR]

Let S be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let I be the determinantal ideal of maximal minors or Pf the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu–Weyman–Witt [Adv. Math. 250 (2014), pp. 596–610] to determine the S-module structures of Ext^j_S(S/I^t, S) and Ext^j_S(S/Pf^t, S), from which we get the degrees of generators of these Ext modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang [J. Pure Appl. Algebra 225 (2021), Paper No. 106789] on the socle degrees of local cohomology modules of the form H^j_\mathfrak {m}(S/I^t). [ABSTRACT FROM AUTHOR]

We provide an axiomatic foundation for a choice model with two periods between which preferences are updated, but the second period choices are positively correlated with past choices in a manner that is unrelated to the agent's preferences. Specifically, in our model, the agent chooses alternative x over alternative y in contrast to his past choice if and only if the difference between the utility of x and that of y is higher than some fixed cost. While restrictive in its nature, this representation captures several related but distinctive phenomena: a taste for consistency, cognitive dissonance, the escalation of commitment, passive choice, and habit formation. We also provide a representation that allows for a more general form of cost and the revealed preference implications of our models. Finally, we connect our representation to the theories of imperfect discrimination. • This paper provides a general treatment for the "positively correlated choice" phenomena. • We provide the first axiomatic foundation for the cognitive dissonance phenomenon over easily observable primitives. • The model's high explanatory and predictive power is demonstrated, and it revealed preference provided. • The paper generalized existing models of guilt and imperfect discrimination. [ABSTRACT FROM AUTHOR]

The rook monoid, also known as the symmetric inverse monoid, is the archetypal structure when it comes to extend the principle of symmetry. In this paper, we establish a Schur–Weyl duality between this monoid and an extension of the classical Schur algebra, which we name the extended Schur algebra. We also explain how this relates to Solomon's Schur–Weyl duality between the rook monoid and the general linear group and mention some advantages of our approach. [ABSTRACT FROM AUTHOR]

This paper studies the structure of semantic theories over modular computational systems and applies the algebraic Theory of Institutions to provide a logical representation of such theories. A modular semantic theory is here defined by a cluster of semantic theories, each for a single program's module, and by a set of relations connecting models of different semantic theories. A semantic theory of a single module is provided in terms of the set of ∑‐models mapped from the category Th of ∑‐theories and generating a hierarchy of structures from an abstract model to a concrete model of data. The collection of abstract models representing different modules of a program is formalised as the category of institutions INS, where theory morphisms express refinements, integrations, and compositions between couples of modules. Finally, it is required that a morphism in INS at any level occurs iff the same morphism occurs at the lower level alongside the Th hierarchy. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic p ≥ 0 , let G be a simple simply connected classical linear algebraic group of rank l and let T be a maximal torus in G with rational character group X (T) . For a nonzero p-restricted dominant weight λ ∈ X (T) , let V be the associated irreducible kG-module. We define ν G (V) as the minimum codimension of any eigenspace on V for any non-central element of G. In this paper, we determine lower-bounds for ν G (V) for G of type A l and dim (V) ≤ l 3 2 , and for G of type B l , C l , or D l and dim (V) ≤ 4 l 3 . Moreover, we give the exact value of ν G (V) for G of type A l with l ≥ 15 ; for G of type B l or C l with l ≥ 14 ; and for G of type D l with l ≥ 16 . [ABSTRACT FROM AUTHOR]

In this paper, we present the connection of two concepts as induced representation and partially reduced irreducible representations (PRIR) appear in the context of port-based teleportation protocols. Namely, for a given finite group G with arbitrary subgroup H, we consider a particular case of matrix irreducible representations, whose restriction to the subgroup H, as a matrix representation of H, is completely reduced to diagonal block form with an irreducible representation of H in the blocks. The basic properties of such representations are given. Then as an application of this concept, we show that the spectrum of the port-based teleportation operator acting on n systems is connected in a very simple way with the spectrum of the corresponding Jucys–Murphy operator for the symmetric group S (n - 1) ⊂ S (n) . This shows on the technical level relation between teleporation and one of the basic objects from the point of view of the representation theory of the symmetric group. This shows a deep connection between the central object describing properties of deterministic PBT schemes and objects appearing naturally in the abstract representation theory of the symmetric group. In particular, we present a new expression for the eigenvalues of the Jucys–Murphy operators based on the irreducible characters of the symmetric group. As an additional but not trivial result, we give also purely matrix proof of the Frobenius reciprocity theorem for characters with explicit construction of the unitary matrix that realizes the reduction in the natural basis of induced representation to the reduced one. [ABSTRACT FROM AUTHOR]

We introduce a new class of quiver varieties, called saw (quiver) varieties. These can be seen as a generalization of the three existing theories: the quiver gauge theory of monopoles, Kac's representation theory of Dynkin types, and Donaldson's theory on the based maps of P 1. E.g., in our terminology, the handsaw (quiver) varieties and chainsaw varieties are finite and affine A type saw varieties respectively with the maximal symmetry breaking. These varieties are the monopole spaces and the caloron spaces in 3d gauge theory, respectively. We expand the theory over to the DE type quiver gauge theory. The saw varieties are constructed via a Hamiltonian formalism, so that the moment maps defining these varieties naturally arise. The main assertion of the paper is that the moment map μ defining a saw variety is a flat morphism for arbitrary dimension vector, if and only if the underlying graph is of finite ADE types. Furthermore under the assumption of maximal symmetry breaking or all the higher frame dimension vectors, the affine ADE types also attain the flatness property of μ. We give an algebro-geometric description of the momentum-zero scheme μ − 1 (0) and the saw variety defined as the Hamiltonian reduction. Various partition functions given rise to by the above mentioned theories are the byproducts of the geometric structure of the corresponding saw varieties. [ABSTRACT FROM AUTHOR]

Consider the n! different unitary matrices that permute nd-dimensional quantum systems. If d ≥ n then they are linearly independent. This paper discusses a sense in which they are approximately orthogonal (with respect to the Hilbert–Schmidt inner product, ⟨ A , B ⟩ = tr A † B / tr I ) if d ≫ n 2 , or, in a different sense, if d ≫ n . Previous work had shown pairwise approximate orthogonality of these matrices, but here we show a more collective statement, quantified in terms of the operator norm distance of the Gram matrix to the identity matrix. This simple point has several applications in quantum information and random matrix theory: (1) showing that random maximally entangled states resemble fully random states, (2) showing that Boson sampling output probabilities resemble those from Gaussian matrices, (3) improving the Eggeling–Werner scheme for multipartite data hiding, (4) proving that the product test of Harrow–Montanaro cannot be performed using LOCC without a large number of copies of the state to be tested, (5) proving that the purity of a quantum state also cannot be efficiently tested using LOCC, and (6, published separately with Brandão and Horodecki) helping prove that poly-size random quantum circuits are poly-designs. [ABSTRACT FROM AUTHOR]

The Frobenius-Perron dimension of a matrix, also known as the spectral radius, is a useful tool for studying linear algebras and plays an important role in the classification of the representation categories of algebras. In this paper, we study the Frobenius-Perron theory of the representation categories of bound quiver algebras containing loops, and find a way to calculate the Frobenius-Perron dimensions of these algebras satisfying the commutativity condition of loops. As an application, we prove that the Frobenius-Perron dimension of the representation category of a modified ADE bounded quiver algebra is equal to the maximal number of loops at each vertex. Finally, we point out that there also exist infinite dimensional algebras whose Frobenius-Perron dimensions is equal to the maximal number of loops by giving an example. [ABSTRACT FROM AUTHOR]

The Hilbert scheme of points |$\textrm {Hilb}^{n}(S)$| of a smooth surface |$S$| is a well-studied parameter space, lying at the interface of algebraic geometry, commutative algebra, representation theory, combinatorics, and mathematical physics. The foundational result is a classical theorem of Fogarty, stating that |$\textrm {Hilb}^{n}(S)$| is a smooth variety of dimension |$2n$|⁠. In recent years there has been growing interest in a natural generalization of |$\textrm {Hilb}^{n}(S)$|⁠ , the nested Hilbert scheme |$\textrm {Hilb}^{(n_{1}, n_{2})}(S)$|⁠ , which parametrizes nested pairs of zero-dimensional subschemes |$Z_{1} \supseteq Z_{2}$| of |$S$| with |$\deg Z_{i}=n_{i}$|⁠. In contrast to Fogarty's theorem, |$\textrm {Hilb}^{(n_{1}, n_{2})}(S)$| is almost always singular, and very little is known about its singularities. In this paper, we aim to advance the knowledge of the geometry of these nested Hilbert schemes. Work by Fogarty in the 70's shows that |$\textrm {Hilb}^{(n,1)}(S)$| is a normal Cohen–Macaulay variety, and Song more recently proved that it has rational singularities. In our main result, we prove that the nested Hilbert scheme |$\textrm {Hilb}^{(n,2)}(S)$| has rational singularities. We employ an array of tools from commutative algebra to prove this theorem. Using Gröbner bases, we establish a connection between |$\textrm {Hilb}^{(n,2)}(S)$| and a certain variety of matrices with an action of the general linear group. This variety of matrices plays a central role in our work, and we analyze it by various algebraic techniques, including the Kempf–Lascoux–Weyman technique of calculating syzygies, square-free Gröbner degenerations, and the Stanley–Reisner correspondence. Along the way, we also obtain results on classes of irreducible and reducible nested Hilbert schemes, dimension of singular loci, and |$F$| -singularities in positive characteristic. [ABSTRACT FROM AUTHOR]

This review is devoted to one of the most interesting and actively developing fields in condensed matter physics—theory of topological insulators. Apart from its importance for theoretical physics, this theory enjoys numerous connections with modern mathematics, in particular, with topology and homotopy theory, Clifford algebras, K-theory and non-commutative geometry. From the physical point of view topological invariance is equivalent to adiabatic stability. Topological insulators are characterized by the broad energy gap, stable under small deformations, which motivates application of topological methods. A key role in the study of topological objects in the solid state physics is played by their symmetry groups. There are three main types of symmetries—time reversion symmetry, preservation of the number of particles (charge symmetry) and PH-symmetry (particle-hole symmetry). Based on the study of symmetry groups and representation theory of Clifford algebras Kitaev proposed a classification of topological objects in solid state physics. In this review we pay special attention to the topological insulators invariant under time reversion. [ABSTRACT FROM AUTHOR]

Although prior research has implied that emotional contagion occurs automatically and unconsciously, convincing evidence suggests that it is significantly influenced by individuals' perceptions of their relationships with others or with collectives within specific social contexts. This implies a role for self-representation in the process. The present study aimed to offer a novel explanation of the interplay between social contexts and emotional contagion, focusing on the contextualized nature of self-representation and exploring the social factors that shape emotional contagion. It further posits a causal loop among social contexts, self-representation, and emotional contagion. Drawing from the lens of self-representation, this study concludes with a discussion on potential research directions in this field, commencing with an exploration of the antecedents and consequences of emotional contagion and selfrepresentation. [ABSTRACT FROM AUTHOR]

The conflict surrounding Islam has been going on for many years, even before the September 11th, 2001 (henceforth, 9/11) incident occurred. However, the fact that the post-incident living circumstances for Muslim people especially in America has been undeniably rough, raises a new trend where Islam is being represented poorly and almost always talked about in negative ways. There are certain characteristics which are attributed especially to identify Muslims, and also certain role that they play in media. Using the Representation theory proposed by Stuart Hall, this research aims to uncover how Islam is represented in American TV series, FBI, which might be seen as a reflection of how Muslims are living in the United States post-9/11 incident. The result found that in the FBI series, Islam is still mainly represented as Arab people, though in reality not every Arabs are Muslims, but the point of view has been slightly shifted for the better. The findings highlight the changing dynamics of Muslim representation, driven by the evolving Muslim community and market demands, contributing to a more nuanced understanding of cultural shifts in the representation of Islam. [ABSTRACT FROM AUTHOR]

Cancellation conditions play a central role in the representation theory of measurement for a weak order on a finite two-dimensional Cartesian product set X. A weak order has an additive representation if and only if it violates no cancellation conditions. Given X, a longstanding open problem is to determine the simplest set of cancellation conditions that is violated by every linear order that is not additively representable. Here, we report that the simplest set of cancellation conditions on a 5 by 5 product X is obtained. [ABSTRACT FROM AUTHOR]

We construct the basic representation of the double affine Hecke algebra at critical level q = 1 associated to an irreducible reduced affine root system R with a reduced gradient root system. For R of untwisted type such a representation was studied by Oblomkov [A. Oblomkov, Double affine Hecke algebras and Calogero–Moser spaces, Represent. Theory 8(2004) 243–266] and further detailed by Gehles [K. E. Gehles, Properties of Cherednik algebras and graded Hecke algebras, PhD thesis, University of Glasgow (2006)] in the presence of minuscule weights. [ABSTRACT FROM AUTHOR]

We prove that, over an arbitrary field, pointwise finite-dimensional persistence modules indexed by 1 decompose uniquely, up to isomorphism, into the direct sum of a bar code and finitely-many Jordan cells. In the language of representation theory, this is a direct sum of string modules and band modules. Persistence modules indexed on 1 have also been called angle-valued or circular persistence modules. We allow either a cyclic order or partial order on 1 and do not have additional finiteness requirements on the modules. We also show that a pointwise finite-dimensional 1 persistence module is indecomposable if and only if it is a bar or Jordan cell. Along the way we classify the isomorphism classes of such indecomposable modules. [ABSTRACT FROM AUTHOR]

In this paper, we study Nijenhuis operators on Leibniz algebras. We discuss the relationship of Nijenhuis operators with Rota-Baxter operators and modified Rota-Baxter operators on Leibniz algebras. We define a representation theory of Nijenhuis Leibniz algebras and construct a cohomology theory. Next, we define a one-parameter formal deformation theory of Nijenhuis Leibniz algebras and study infinitesimals, rigidity, and equivalences along the line of Gerstenhaber deformation theory. As applications of our cohomology theory, we show that our cohomology is deformation cohomology and study abelian extensions of such algebras. [ABSTRACT FROM AUTHOR]

Integrality properties of partial sums over irreducible representations, along columns of character tables of finite groups, were recently derived using combinatorial topological string theories (CTST). These CTST were based on Dijkgraaf-Witten theories of flat G -bundles for finite groups G in two dimensions, denoted G -TQFTs. We define analogous combinatorial topological strings related to two dimensional topological field theories (TQFTs) based on fusion coefficients of finite groups. These TQFTs are denoted as R (G)-TQFTs and allow analogous integrality results to be derived for partial row sums of characters over conjugacy classes along fixed rows. This relation between the G -TQFTs and R (G)-TQFTs defines a row-column duality for character tables, which provides a physical framework for exploring the mathematical analogies between rows and columns of character tables. These constructive proofs of integrality are complemented with the proof of similar and complementary results using the more traditional Galois theoretic framework for integrality properties of character tables. The partial row and column sums are used to define generalised partitions of the integer row and column sums, which are of interest in combinatorial representation theory. [ABSTRACT FROM AUTHOR]

We study the simplest example of mirror symmetry for 3d = 4 SUSY gauge theories: the A-twist of a free hypermultiplet and the B-twist of SQED. We particularly focus on the category of line operators in each theory. Using the work of Costello–Gaiotto, we define these categories as appropriate categories of modules for the boundary vertex operator algebras present in each theory. For the A-twist of a free hyper, this will be a certain category of modules for the β γ VOA, properly containing the category previously studied by Allen-Wood. Applying the work of Creutzig–Kanade–McRae and Creutzig–McRae–Yang, we show that the category of line operators on the A side possesses the structure of a braided tensor category, extending the result of Allen-Wood. In addition, we prove that there is a braided tensor equivalence between the categories of line operators on the A side and B side, completing a nontrivial check of the 3d mirror symmetry conjecture. We derive explicit fusion rules as a consequence of this equivalence and obtain interesting relations with associated quantum group representations. [ABSTRACT FROM AUTHOR]

This study investigates the scaling behavior of the ground-state entanglement entropy in a model of free fermions on folded cubes. An analytical expression is derived in the large-diameter limit, revealing a strict adherence to the area law. The absence of the logarithmic enhancement expected for free fermions is explained using a decomposition of folded cubes in chains based on its Terwilliger algebra and s o (3) − 1 . The entanglement Hamiltonian and its relation to Heun operators are also investigated. [ABSTRACT FROM AUTHOR]

Grigoriev (2001) and Laurent (2003) independently showed that the sum-of-squares hierarchy of semidefinite programs does not exactly represent the hypercube {+¹} n until degree at least n of the hierarchy. Laurent also observed that the pseudomoment matrices her proof constructs appear to have surprisingly simple and recursively structured spectra as n increases. While several new proofs of the Grigoriev-Laurent lower bound have since appeared, Laurent's observations have remained unproved. We give yet another, representation-theoretic proof of the lower bound, which also yields exact formulas for the eigenvalues of the Grigoriev--Laurent pseudomoments. Using these, we prove and elaborate on Laurent's observations. Our proof shows that the Grigoriev-Laurent pseudomoments are a special case of a Gram matrix construction of pseudomoments proposed by Bandeira and Kunisky (2020). In the course of the proof, we also find a new realization of the irreducible representations of the symmetric group corresponding to Young diagrams with two rows, as spaces of multivariate polynomials that are multiharmonic with respect to an equilateral simplex. [ABSTRACT FROM AUTHOR]