Related papers: Recent Progress in the Symmetric Generation of Gro…
We give a new characterization of partial groups as a subcategory of symmetric (simplicial) sets. This subcategory has an explicit reflection, which permits one to compute colimits in the category of partial groups. We also introduce the…
The convolution of indicators of two conjugacy classes on the symmetric group S_q is usually a complicated linear combination of indicators of many conjugacy classes. Similarly, a product of the moments of the Jucys--Murphy element involves…
It is well known that every finite simple group has a generating pair. Moreover, Guralnick and Kantor proved that every finite simple group has the stronger property, known as $\frac{3}{2}$-generation, that every nontrivial element is…
We construct an explicit generating sets $F_n$ and $\tilde F_n$ of the alternating and the symmetric groups, which make the Cayley graphs $C(Alt(n), F_n)$ and $C(Sym(n), \tilde F_n)$ a family of bounded degree expanders for all sufficiently…
The background of this paper is the following: search of the minimal systems of generators for this class of group which still was not founded also problem of representation for this class of group, exploration of systems of generators for…
This survey paper concerns mainly with some asymptotic topological properties of finitely presented discrete groups: quasi-simple filtration (QSF), geometric simple connectivity (GSC), topological inverse-representations, and the notion of…
This paper introduces the concept of a generating set for stochastic matrices -- a subset of matrices whose repeated composition generates the entire set. Understanding such generating sets requires specifying the "indivisible elements" and…
The emergence of generative AI models has dramatically expanded the availability and use of synthetic data across scientific, industrial, and policy domains. While these developments open new possibilities for data analysis, they also raise…
A totally symmetric set is a finite subset of a group for which any permutation of the elements can be realized by conjugation in the ambient group. Such sets are rigid under homomorphisms, and so exert a great deal of control over the…
This work presents the first statistical performance guarantees for group-invariant generative models. Many real data, such as images and molecules, are invariant to certain group symmetries, which can be taken advantage of to learn more…
This survey paper aims at giving the state of the art in the study of string C-group representations of almost simple groups. It also suggest a series of problems and conjectures to the interested reader.
We introduce a new combinatorial structure: the superselector. We show that superselectors subsume several important combinatorial structures used in the past few years to solve problems in group testing, compressed sensing, multi-channel…
It was shown that in a group of bijections of an infinite set some families of subsets, related to the cardinality of some eigenspaces, are generating. Besides, we derived a criterion for generating by sets of this kind.
Symmetric homology is an analog of cyclic homology in which the cyclic groups are replaced by symmetric groups. The foundations for the theory of symmetric homology of algebras are developed in the context of crossed simplicial groups using…
The mapping class group of an orientable surface, which records its symmetries up to isotopy, plays a central role in low-dimensional topology. This chapter explores the foundational problem of determining minimal generating sets for these…
Mutually unbiased bases that can be cyclically generated by a single unitary operator are of special interest, since they can be readily implemented in practice. We show that, for a system of qubits, finding such a generator can be cast as…
Let $G$ be a group. Then $S\subseteq G$ is an invariable generating set of $G$ if every subset $S'$ obtained from $S$ by replacing each element with a conjugate is also a generating set of $G$. We investigate invariable generation among key…
Symmetry plays a fundamental role in understanding natural phenomena and mathematical structures. This work develops a comprehensive theory for studying the persistent symmetries and degree of asymmetry of finite point configurations over…
In this paper, we investigate the structure of associated groups of symmetric quandles. Among other results, we explore the relationship between the associated group of a symmetric quandle and that of its underlying quandle. We provide a…
Many systems of interest exhibit nested emergent layers with their own rules and regularities, and our knowledge about them seems naturally organised around these levels. This paper proposes that this type of hierarchical emergence arises…