Characters of infinite-dimensional quantum classical groups: BCD cases

We study the character theory of inductive limits of [Formula: see text]-deformed classical compact groups. In particular, we clarify the relationship between the representation theory of Drinfeld–Jimbo quantized universal enveloping algebras and our previous work on the quantized characters. We also apply the character theory to construct Markov semigroups on unitary duals of [Formula: see text], [Formula: see text], and their inductive limits.

Triangular matrix coalgebras: Representation theory and recollement

For any triangular matrix coalgebra [Formula: see text], in this paper, we first examine some connections between coalgebra properties of [Formula: see text] and its constituent coalgebras [Formula: see text], [Formula: see text], which contain semiperfectness, computability and row/column-finiteness of their left Cartan matices. Then we devote to considering the coresolution dimensions of recollement of comodule categories by investigating covariantly finite subcategories.

Recursive structure of the Gauß hypergeometric function and boundary/crosscap conformal block

The Gauß hypergeometric function shows a recursive structure which resembles those found in conformal blocks. We compare it with the recursive structure of the conformal block in boundary/crosscap conformal field theories that is obtained from the representation theory. We find that the pole structure perfectly agrees but the recursive structure in the Gauß hypergeometric function is slightly “off-shell”.

Going beyond Cryptocurrencies: The Social Representation of Blockchain

This study investigates the social representation of Blockchain from the perspective of professionals in Brazil, herein considered as a proxy for emerging markets, and then compares the results found with the existing academic literature on the concept of Blockchain. To do that, the social representation theory was applied, operationalized through the words evocation technique. Security, bitcoin and decentralization were the categories located in the central nucleus of the social representation of Blockchain, while innovation, data, network, cryptocurrency, and technology were the categories located in the peripheral system. Based on the results obtained, there was a perceived strong association of Blockchain with bitcoin, one of its applications, and a dissonance between the existing academic literature and the perception of Brazilian professionals about the concept of Blockchain, as the latter is a privilege of the technical and operational issues of Blockchain to the detriment of its strategic potential. This dissonance can cause Blockchain initiatives to have results below expectations. Finally, Brazilian professionals did not realize the potential for inclusion of Blockchain in an emerging market such as Brazil and did not notice the need and relevance of a specific legal governance for Blockchain, an issue also forgotten by academia.

Representations in Intergroup Relations: Reflexivity, Meta-Representations, and Interobjectivity

Social and cultural groups are characterised by shared systems of social objects and issues that constitute their objective reality and their members' identity. It is argued that interpersonal interactions within such groups require a system of comprehensive representations to enable concerted interaction between individuals. Comprehensive representations include bits and pieces of the interactant's representational constitution and potential values and behaviours to reduce possible friction in interactions. On a larger scale, the same is true in encounters, communication, and interaction between members of different cultural groups where interactants need to dispose of a rough knowledge of the other culture's relevant characteristics. This mutual knowledge is called meta-representations that complement the actors' own values and ways of thinking. This concept complements Social Representation Theory when applied to cross-cultural and inter-ethnic interactions.

The Representation Theory of Neural Networks

In this work, we show that neural networks can be represented via the mathematical theory of quiver representations. More specifically, we prove that a neural network is a quiver representation with activation functions, a mathematical object that we represent using a network quiver. Furthermore, we show that network quivers gently adapt to common neural network concepts such as fully connected layers, convolution operations, residual connections, batch normalization, pooling operations and even randomly wired neural networks. We show that this mathematical representation is by no means an approximation of what neural networks are as it exactly matches reality. This interpretation is algebraic and can be studied with algebraic methods. We also provide a quiver representation model to understand how a neural network creates representations from the data. We show that a neural network saves the data as quiver representations, and maps it to a geometrical space called the moduli space, which is given in terms of the underlying oriented graph of the network, i.e., its quiver. This results as a consequence of our defined objects and of understanding how the neural network computes a prediction in a combinatorial and algebraic way. Overall, representing neural networks through the quiver representation theory leads to 9 consequences and 4 inquiries for future research that we believe are of great interest to better understand what neural networks are and how they work.

On the representation theory of the vertex algebra L−5/2(sl(4))

We study the representation theory of non-admissible simple affine vertex algebra [Formula: see text]. We determine an explicit formula for the singular vector of conformal weight four in the universal affine vertex algebra [Formula: see text], and show that it generates the maximal ideal in [Formula: see text]. We classify irreducible [Formula: see text]-modules in the category [Formula: see text], and determine the fusion rules between irreducible modules in the category of ordinary modules [Formula: see text]. It turns out that this fusion algebra is isomorphic to the fusion algebra of [Formula: see text]. We also prove that [Formula: see text] is a semi-simple, rigid braided tensor category. In our proofs, we use the notion of collapsing level for the affine [Formula: see text]-algebra, and the properties of conformal embedding [Formula: see text] at level [Formula: see text] from D. Adamovic et al. [Finite vs infinite decompositions in conformal embeddings, Comm. Math. Phys. 348 (2016) 445–473.]. We show that [Formula: see text] is a collapsing level with respect to the subregular nilpotent element [Formula: see text], meaning that the simple quotient of the affine [Formula: see text]-algebra [Formula: see text] is isomorphic to the Heisenberg vertex algebra [Formula: see text]. We prove certain results on vanishing and non-vanishing of cohomology for the quantum Hamiltonian reduction functor [Formula: see text]. It turns out that the properties of [Formula: see text] are more subtle than in the case of minimal reduction.

Kronecker positivity and 2-modular representation theory

This paper consists of two prongs. Firstly, we prove that any Specht module labelled by a 2-separated partition is semisimple and we completely determine its decomposition as a direct sum of graded simple modules. Secondly, we apply these results and other modular representation theoretic techniques on the study of Kronecker coefficients and hence verify Saxl’s conjecture for several large new families of partitions. In particular, we verify Saxl’s conjecture for all irreducible characters of S n \mathfrak {S}_n which are of 2-height zero.

Shadows in the Wild - Folded Galleries and Their Applications

This survey is about combinatorial objects related to reflection groups and their applications in representation theory and arithmetic geometry. Coxeter groups and folded galleries in Coxeter complexes are introduced in detail and illustrated by examples. Further it is explained how they relate to retractions in Bruhat-Tits buildings and to the geometry of affine flag varieties and affine Grassmannians. The goal is to make these topics accessible to a wide audience.