Optimal Line Packings from Nonabelian Groups

We present the theory of equiangular tight frames (ETFs) from association schemes as a generalization of harmonic ETFs. This theory has the pleasing consequence of including not only harmonic ETFs from abelian groups but also real ETFs with centroidal symmetry and the ETFs discovered by Renes and Strohmer coming from skew-symmetric conference matrices. Finally we present a theory of nonabelian harmonic ETFs from the perspective of association schemes. An infinite class of ETFs arising as group frames from nonabelian groups is presented.

This paper proposes a method for constructing an improved encryption scheme on generalized Suzuki 2-groups for the MST3 cryptosystem, which improves the security parameters of the original approach. The challenge of improving existing cryptosystem design approaches is driven by advances in building quantum computers with sufficient computing power to render many public-key cryptosystems insecure. In particular, this includes cryptosystems based on the factorization problem or the discrete logarithm problem, such as RSA and ECC. There have been several proposals in the past two decades for using non-commutative groups to create quantum-resistant cryptosystems. The unsolvable word problem is a promising area of research for building cryptosystems. It was formulated by Wagner and Magyarik and lies in the realm of permutation groups. Magliveras proposed logarithmic signatures, which are a special type of factorization that applies to finite groups. The latest version of this implementation, known as MST3, is based on the Suzuki group. In 2008, Magliveras demonstrated a transitive LS limit for the MST3 cryptosystem. Later, Svaba proposed the eMST3 cryptosystem with improved security parameters, achieved by adding a secret homomorphic cover. In 2018, T. van Trung proposed an MST3 approach using strong aperiodic logarithmic signatures for abelian p-groups. Kong and his colleagues conducted an extensive analysis of MST3 and noted that, since there are currently no publications on the quantum vulnerability of the algorithm, it can be considered a candidate for use in the post-quantum era. The first implementation of the cryptosystem on the generalized Suzuki 2-group does not provide encryption of the entire Suzuki 2-group and does not protect against attacks with sequential key recovery by the brute-force method. Further work has developed the idea of public cryptography using non-Abelian refinements of parameters. This paper proposes a method for constructing an encryption scheme on Suzuki 2-groups that improves the security parameters of the existing MST3 cryptosystem and solves its security problems.

The method for a key agreement protocol (KAP) with application of a non-commutative group is presen-ted. The method is based on the two one-way functions (OWFs) and corresponding hard problems. One hard problem is left right factors' search problem (LRFSP) in non-commutative group presentation level, and another one is a postu-lated hard problem in the non-commutative group representation level or, in other words, action level. On the basis of LRFSP the first one-way function (OWF1) is constructed. Non-commutative group is treated as an endomorphic group of operators acting on a certain semimodule or module. In this case such semimodule is compatible with a non-commutative group and algebra generated by this non-commutative group. The requirements of semimodule compatibility to the introduced algebra are presented. On the basis of a hard problem postulation in non-commutative group action level the second OWF2 is constructed.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

One of the results of the theory of the irreducible representations of the unitary group in n dimensions U n is that these representations, if restricted to the subgroup U n − 1 leaving a vector (let us say the unit vector e l along the first coordinate axis) invariant, do not contain any irreducible representation of this U n − 1 more than once (see [1, Chapter X and Equation (10.21)]; the irreducible representations of the unitary group were first determined by I. Schur in his doctoral dissertation (Berlin, 1901)). Some time ago, a criterion for this situation was derived for finite groups [3] and the purpose of the present article is to prove the aforementioned result for compact Lie groups, and to apply it to the theory of the representations of U n .KeywordsCharacteristic VectorIrreducible RepresentationFinite GroupUnitary GroupGroup VolumeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper gives a survey on the relation between Hibi algebras and representation theory. The notion of Hodge algebras or algebras with straightening laws has been proved to be very useful to describe the structure of many important algebras in classical invariant theory and representation theory (Bruns and Herzog 1993; De Concini et al. 1982; Eisenbud 1980; Gonciulea and Lakshmibai 2001; Seshadri 2007). In particular, a special type of such algebras introduced by Hibi (1987) provides a nice bridge between combinatorics and representation theory of classical groups. We will examine certain poset structures of Young tableaux and affine monoids, Hibi algebras in toric degenerations of flag varieties, and their relations to polynomial representations of the complex general linear group.

This study investigates the representation theory of the multiplicative group of a finite nearfield, with a specific focus on the group <img src=image/13440431_01.gif>, arising from Dickson nearfields. Nearfields, generalizations of fields lacking one of the two distributive laws, pose interesting challenges in algebraic structures due to their noncommutativity and asymmetry. Their multiplicative groups, particularly those derived from Dickson nearfields, have a metacyclic structure that allows for intricate analysis using representation theory. The purpose of this research is to construct and analyze the character tables of these multiplicative groups. We explore foundational concepts such as Schur's lemma, Maschke's theorem, and orthogonality relations, and apply them within the context of finite nearfields. A combination of theoretical and computational methodologies is employed. Specifically, we make use of the Magma algebra system to build and verify character tables for <img src=image/13440431_01.gif> for small values of <img src=image/13440431_02.gif> and <img src=image/13440431_03.gif>, including detailed derivations for <img src=image/13440431_04.gif> and <img src=image/13440431_05.gif>. The principal results include explicit character tables, classifications of conjugacy classes, degrees of irreducible representations, and a generalization framework for any pair (<img src=image/13440431_06.gif>). The research demonstrates that the multiplicative groups <img src=image/13440431_01.gif> are not only metacyclic but also nilpotent and solvable under specific conditions. It further establishes structural conditions that determine the conjugacy class partitioning and representation degrees. Our conclusions highlight the deep connection between nearfield algebraic properties and finite group representation theory. This work contributes to the theoretical development of group representations in non-classical algebraic settings and opens doors to applications in geometry, coding theory, and cryptography. Practically, the use of Magma facilitates automated representation analysis, and socially, this research promotes a broader understanding of non-traditional algebraic systems. While the current study is computationally verified for small instances, future work will aim to optimize algorithms for handling larger <img src=image/13440431_01.gif> structures, enhancing scalability and applicability.

Introduction Representations of $\mathbf {SU}(2)$ The general theory of linear representations of compact groups Lie theory of representations of compact connected Lie groups Induced representations of compact groups Spherical functions on compact groups Examples spherical harmonics The general theory of spherical functions Fourier and Plancherel transforms Extension of the Plancherel transform The subtleties of harmonic analysis Differential properties of spherical functions on Lie groups Spherical functions on semisimple Lie groups More on $\mathbf {SL}(2, \mathbf R)$ Automorphic functions Groups of isometries and Bessel functions Other special functions References.

Difference sets with parameters (\frac{q^{d + 1} - 1}{q - 1}, \frac{q^d - 1}{q - 1}, \frac{q^{d - 1} - 1}{q - 1}), where q is a prime power and d \geq 1, are known to exist in cyclic groups and are called classical Singer difference sets. We study a special case of this family with q = 7 and d = 3 in search of more difference sets. According to GAP, there are 220 groups of order 400 out of which 10 are abelian. E. Kopilovich and other authors showed that the remaining nine abelian groups of order 400 do not admit (400, 57, 8) difference sets. Also, Gao and Wei used the (400, 57, 8) Singer difference set to construct four inequivalent difference sets in a non-abelian group. In this paper, we demonstrate using group representation and factorization in cyclotomic rings that, out of the remaining 209 non-abelian groups of order 400, only 15 could possibly admit (400, 57, 8) difference sets.

We survey recent results regarding embeddings of finite simple groups (and their nonsplit central extensions) in complex Lie groups, especially the Lie groups of exceptional type. Introduction Throughout this paper, L will be a finite group. Representation theory for L is usually understood to be the study of group morphisms L → GL(n, k) for distinguished collections of fields k (e.g., all overfields of a fixed field F ) and positive integers n . The topic of this survey is motivated by the question as to what happens if GL ( n ,·) is replaced by another algebraic group G (·). We shall mainly be concerned with the case where L is a finite simple group (that is, a finite nonabelian simple group) or a central extension thereof, and G(k) is a connected simple algebraic group over a field k . A further restriction of our discussion concerns the field k . It will mostly be taken to be the complex numbers, in which case we will mainly study group morphisms from L to the complex Lie group G (ℂ). (See below for some exceptions in §3 and §5.) For G (·) of classical type, the theory for representations L → G (ℂ) differs little from the usual one for GL ( n , ℂ). Indeed, a representation L → GL ( n , ℂ) decomposes into irreducible subrepresentations. The decomposition is well controlled by character theory.

The theory of infinite dimensional group representations may be regarded as a blend of Fourier analysis and the classical purely algebraic theory of group representations.

Representation theory is concerned with the ways of explaining or visualizing a group as a group of matrices. In this paper, we extend the permutation pattern of to a two-line notation. We consider the representations of this non-deranged permutation group(p ≥ 5 and p a prime). Also we reveal some interesting properties and results of the character of where .

This dissertation is concerned with the interplay between the theory of Banach spaces and representations of groups. The wavelet transform has proven to be a useful tool in characterizing and constructing Banach spaces, and we investigate a generalization of an already known technique due to H.G. Feichtinger and K. Gröchenig. This generalization is presented in Chapter 3, and in Chapters 4 and 5 we present examples of spaces which can be described using the theory. The first example clears up a question regarding a wavelet characterization of Bergman spaces related to a non-integrable representation. The second example is a wavelet characterization of Besov spaces on the forward light cone.

Editorial

Variant [formula omitted] supersymmetric non-Abelian Proca–Stueckelberg formalism in four dimensions