Related papers: Frame potential and finite abelian groups

This paper concentrates on positive definite functions on finite abelian groups, which are central to harmonic analysis and related fields. By leveraging the group structure and employing Fourier analysis, we establish a lower bound for the…

We study the growth of torsion in the abelianizations of finite index subgroups in finitely generated metabelian groups. This complements earlier work of Kar, Kropholler and the author which covered the finitely presented amenable groups.

Motivated by the recent work of Bownik and Ross \cite{BR}, and Jakobsen and Lemvig \cite{JL}, this article generalizes latest results on reproducing formulas for generalized translation invariant (GTI) systems to the setting of super-spaces…

This paper concerns the overcompleteness of coherent frames for unimodular amenable groups. It is shown that for coherent frames associated with a localized vector a set of positive Beurling density can be removed yet still leave a frame.…

This is an introduction to the finite groups, with focus on the groups of permutations and reflections, and more generally, on the finite groups of unitary matrices. We first discuss the basics of group theory, featuring the cyclic,…

We introduce a class of finite tight frames called prime tight frames and prove some of their elementary properties. In particular, we show that any finite tight frame can be written as a union of prime tight frames. We then characterize…

It is a simple fact that a subgroup generated by a subset $A$ of an abelian group is the direct sum of the cyclic groups $\langle a\rangle$, $a\in A$ if and only if the set $A$ is independent. In [5] the concept of an $independent$ set in…

Motivated by some known problems concerning combinatorial structures associated with finite one-dimensional affine permutation groups, we study subgroups which are closed in $\operatorname{\Gamma{L}}_1(q)$. This brings us to a description…

Totally symmetric sets are a recently introduced tool for studying homomorphisms between groups. In this paper, we give full classifications of totally symmetric sets in certain families of groups and bound their sizes in others. As a…

I investigate modal group theory for arbitrary homomorphisms. Possibility is interpreted by the existence of a group homomorphism out of the given group, so the semantics is governed by the possibility of collapse: elements may be…

Let $K$ be a compact group, and let $\rho$ be a representation of $K$ on a Hilbert space $\mathcal{H}_\rho$. We classify invariant subspaces of $\mathcal{H}_\rho$ in terms of range functions, and investigate frames of the form $\{\rho(\xi)…

Our main objective is to show that the computational methods that we previously developed to search for difference families in cyclic groups can be fully extended to the more general case of arbitrary finite abelian groups. In particular…

We investigate the conditions for a finite abelian group $G$ under which any cyclic subgroup $H$ and any group homomorphism $f \in \operatorname{Hom}(H,G)$ can be extended to an endomorphism $F \in \operatorname{End}(G)$. As a result, we…

We continue classification of finite groups which can be used as symmetry group of the scalar sector of the four-Higgs-doublet model (4HDM). Our objective is to systematically construct non-abelian groups via the group extension procedure,…

We classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules. One application is a proof of the telescope conjecture in this context.…

Frame theory has been a popular subject in the design of structured signals and codes in recent years, with applications ranging from the design of measurement matrices in compressive sensing, to spherical codes for data compression and…

We prove the existence of tight frames whose elements lie on an arbitrary ellipsoidal surface within a real or complex separable Hilbert space H, and we analyze the set of attainable frame bounds. In the case where H is real and has finite…

Degree bounds for algebra generators of invariant rings are a topic of longstanding interest in invariant theory. We study the analogous question for field generators for the field of rational invariants of a representation of a finite…

Classical and recent results on uncertainty principles for functions on finite Abelian groups relate the cardinality of the support of a function to the cardinality of the support of its Fourier transforms. We use these results and their…

It is well-known that characters classify linear representations of finite groups, that is if characters of two representations of a finite group are the same, these representations are equivalent. It is also well-known that, in general,…