Representations Of Algebraic Groups Research Articles

Multiplier systems for Siegel modular groups

Deligne proved in [Extensions centrales non résiduellement finies de groupes arithmetiques, C. R. Acad. Sci. Paris 287 (1978) 203–208] (see also 7.1 in [R. Hill, Fractional weights and non-congruence subgroups, in Automorphic Forms and Representations of Algebraic Groups Over Local Fields, eds. H. Saito and T. Takahashi, Surikenkoukyuroku Series, Vol. 1338 (2003), pp. 71–80]) that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus [Formula: see text] must be integral or half integral. Actually he proved that for a system [Formula: see text] of complex numbers of absolute value 1 [Formula: see text] can be an automorphy factor only if [Formula: see text] is integral. We give a different proof for this. It uses Mennicke’s result [Zur Theorie der Siegelschen Modulgruppe, Math. Ann. 159 (1965) 115–129] that subgroups of finite index of the Siegel modular group are congruence subgroups and some techniques from [Solution of the congruence subgroup problem for [Formula: see text] and [Formula: see text], Publ. Math. Inst. Hautes Études Sci. 33 (1967) 59–137] of Bass–Milnor–Serre.

On abelian unitary involutions of crossed products

In the theory of classical linear algebraic groups, of importance are special unitary groups of non-commuted involution crossed products with division. The description of these groups largely depends on the involution type of these products. The class of Abelian involution crossed products is considered and the criterion for their existence is set provided that unitary bases (with respect to these involutions) are present in these products.

Technology of protection of authentication data of computer network users

The article is devoted to the problem of perfect protection of authentication data of users of computer systems, especially in the case of a large number of different rights and powers that are personally granted to users. The importance of this problem increases especially when the number of such users in the system is hundreds or thousands. At the same time, each of them must create conditions for access only to their data and ensure protection against any outside interference, both by other users and by staff. A typical example of such conditions are secret electronic voting systems. The software and technical solutions described in this work have passed many years of testing and continue to be practically used in the electronic voting system of the Kyiv National University of Civil Engineering and Architecture. This system is regularly used for conducting elections to student self-government bodies and for surveys among students regarding the quality of teaching subjects. With the help of this system, elections of leaders of the Red Cross Society of Ukraine were held during the pandemic, which was associated with the Covid-19 virus. The mathematical basis of the described authentication data protection technology is the theory of algebraic groups, namely, the problem of discrete logarithmization over large Galois fields. Thanks to the use of cryptographic transformations over these fields, instead of known hash functions, it was possible to get rid of the possibility of collisions and neutralize the disclosure of passwords by attackers who were able to do it with the help of specialized Internet resources. It was also possible to increase resistance to brute-force password cracking, since conversions to Galois fields take ten times more time than hash function calculations.

Multiplicity-free Representations of Algebraic Groups

Let K K be an algebraically closed field of characteristic zero, and let G G be a connected reductive algebraic group over K K . We address the problem of classifying triples ( G , H , V ) (G,H,V) , where H H is a proper connected subgroup of G G , and V V is a finite-dimensional irreducible G G -module such that the restriction of V V to H H is multiplicity-free – that is, each of its composition factors appears with multiplicity 1. A great deal of classical work, going back to Dynkin, Howe, Kac, Stembridge, Weyl and others, and also more recent work of the authors, can be set in this context. In this paper we determine all such triples in the case where H H and G G are both simple algebraic groups of type A A , and H H is embedded irreducibly in G G . While there are a number of interesting familes of such triples ( G , H , V ) (G,H,V) , the possibilities for the highest weights of the representations defining the embeddings H > G H>G and G > G L ( V ) G>GL(V) are very restricted. For example, apart from two exceptional cases, both weights can only have support on at most two fundamental weights; and in many of the examples, one or other of the weights corresponds to the alternating or symmetric square of the natural module for either G G or H H .

St Petersburg school of linear groups. II. Early works by Suslin

The present survey describes the contribution of St Petersburg mathematicians to the theory of linear, classical and algebraic groups. The second part is devoted to the publications by Suslin of the 1970s and the early 1980s, in the areas of classical algebraic K-theory and the theories of linear and classical groups. Also, we describe the general context of these works, state some of the most important results by Suslin himself, and his students, and some of the most closely related follow-ups and subsequent results.

Research on the group theory based on the perspective of the Rubiks Cube

With the development of mathematics, more and more fields of study have been created and progressed, it is worth to implement the knowledge into the real life. There is a well-known puzzle called the Rubiks Cube, has many connections to a branch of abstract algebra group theory. Therefore, this paper will discuss how the Rubiks Cube showing the properties from group theory, by introducing basic knowledges of group theory, followed by examples in terms of this intelligent toy. This paper will first introduce the properties of the Rubiks Cube, then move to the construction of its group. Subsequently, the four axioms that form a group are explained. After that, the reasons why the operations of the Rubiks Cube are able to form a group are explained as the examples of those four axioms. It is followed by the concepts in group theory, and provisions of the exemplifications in terms of the Rubiks Cube, such as closure, cyclicity, Cayleys graph. Explaining the group theory from the perspective of the Rubiks Cube provides a tangible channel to learn the intangible knowledges effectively. Learners are able to study these hard knowledges easily by rotating a simple toy and observing the conclusions.

Constructions of difference sets in nonabelian 2-groups

Difference sets have been studied for more than 80 years. Techniques from algebraic number theory, group theory, finite geometry, and digital communications engineering have been used to establish constructive and nonexistence results. We provide a new theoretical approach which dramatically expands the class of $2$-groups known to contain a difference set, by refining the concept of covering extended building sets introduced by Davis and Jedwab in 1997. We then describe how product constructions and other methods can be used to construct difference sets in some of the remaining $2$-groups. We announce the completion of ten years of collaborative work to determine precisely which of the 56,092 nonisomorphic groups of order 256 contain a difference set. All groups of order 256 not excluded by the two classical nonexistence criteria are found to contain a difference set, in agreement with previous findings for groups of order 4, 16, and 64. We provide suggestions for how the existence question for difference sets in $2$-groups of all orders might be resolved.

Multiple Multi-Orbit Pairing Algebras in Nuclei

The algebraic group theory approach to pairing in nuclei is an old subject and yet it continues to be important in nuclear structure, giving new results. It is well known that for identical nucleons in the shell model approach with j − j coupling, pairing algebra is SU(2) with a complementary number-conserving Sp(N) algebra and for nucleons with good isospin, it is SO(5) with a complementary number-conserving Sp(2Ω) algebra. Similarly, with L − S coupling and isospin, the pairing algebra is SO(8). On the other hand, in the interacting boson models of nuclei, with identical bosons (IBM-1) the pairing algebra is SU(1, 1) with a complementary number-conserving SO(N) algebra and for the proton–neutron interacting boson model (IBM-2) with good F-spin, it is SO(3, 2) with a complementary number-conserving SO(ΩB) algebra. Furthermore, in IBM-3 and IBM-4 models several pairing algebras are possible. With more than one j or ℓ orbit in shell model, i.e., in the multi-orbit situation, the pairing algebras are not unique and we have the new paradigm of multiple pairing [SU(2), SO(5) and SO(8)] algebras in shell models and similarly there are multiple pairing algebras [SU(1, 1), SO(3, 2) etc.] in interacting boson models. A review of the results for multiple multi-orbit pairing algebras in shell models and interacting boson models is presented in this article with details given for multiple SU(2), SO(5), SU(1, 1) and SO(3, 2) pairing algebras. Some applications of these multiple pairing algebras are discussed. Finally, multiple SO(8) pairing algebras in shell model and pairing algebras in IBM-3 model are briefly discussed.

St Petersburg school of linear groups. I. Prehistorical period

The present survey describes the contribution of St Petersburg mathematicians to the theory of linear, classical and algebraic groups. The first part is dedicated to the prehistorical period, the historical genesis of Tartakowski and Faddeev algebra schools, and to the general outline of the works by Borewicz and Suslin of the mid-1970s that initiated systematical research in the fields of classical groups and algebraic K-theory in St Petersburg.

Galois/Monodromy Groups for Decomposing Minimal Problems in 3D Reconstruction

We consider Galois/monodromy groups arising in computer vision applications, with a view towards building more efficient polynomial solvers. The Galois/monodromy group allows us to decide when a given problem decomposes into algebraic subproblems, and whether or not it has any symmetries. Tools from numerical algebraic geometry and computational group theory allow us to apply this framework to classical and novel reconstruction problems. We consider three classical cases--3-point absolute pose, 5-point relative pose, and 4-point homography estimation for calibrated cameras--where the decomposition and symmetries may be naturally understood in terms of the Galois/monodromy group. We then show how our framework can be applied to novel problems from absolute and relative pose estimation. For instance, we discover new symmetries for absolute pose problems involving mixtures of point and line features. We also describe a problem of estimating a pair of calibrated homographies between three images. For this problem of degree 64, we can reduce the degree to 16; the latter better reflecting the intrinsic difficulty of algebraically solving the problem. As a byproduct, we obtain new constraints on compatible homographies, which may be of independent interest.

Jordan Types of Triangular Matrices over a Finite Field

Let \(\lambda \) be a partition of an integer n and \({\mathbb F}_q\) be a finite field of order q. Let \(P_\lambda (q)\) be the number of strictly upper triangular \(n\times n\) matrices of the Jordan type \(\lambda \). It is known that the polynomial \(P_\lambda \) has a tendency to be divisible by high powers of q and \(Q=q-1\), and we put \(P_\lambda (q)=q^{d(\lambda )}Q^{e(\lambda )}R_\lambda (q)\), where \(R_\lambda (0)\ne 0\) and \(R_\lambda (1)\ne 0\). In this article, we study the polynomials \(P_\lambda (q)\) and \(R_\lambda (q)\). Our main results: an explicit formula for \(d(\lambda )\) (an explicit formula for \(e(\lambda )\) is known, see Proposition 3.3 below), a recursive formula for \(R_\lambda (q)\) (a similar formula for \(P_\lambda (q)\) is known, see Proposition 3.1 below), the stabilization of \(R_\lambda \) with respect to extending \(\lambda \) by adding strings of 1’s, and an explicit formula for the limit series \(R_{\lambda 1^\infty }\). Our studies are motivated by projected applications to the orbit method in the representation theory of nilpotent algebraic groups over finite fields.

Structural position vectors and symmetries in complex networks.

Symmetries, due to their fundamental importance to dynamical processes on networks, have attracted a great deal of current research. Finding all symmetric nodes in large complex networks typically relies on automorphism groups from algebraic-group theory, which are solvable in quasipolynomial time. We articulate a conceptually appealing and computationally extremely efficient approach to finding and characterizing all symmetric nodes by introducing a structural position vector (SPV) for each node in networks. We establish the mathematical result that symmetric nodes must have the same SPV value and demonstrate, using six representative complex networks from the real world, that all symmetric nodes in these networks can be found in linear time. Furthermore, the SPVs not only characterize the similarity of nodes but also quantify the nodal influences in propagation dynamics. A caveat is that the proved mathematical result relating the SPV values to nodal symmetries is not sufficient; i.e., nodes having the same SPV values may not be symmetric, which arises in regular networks or networks with a dominant regular component. We point out with an analysis that this caveat is, in fact, shared by the known existing approaches to finding symmetric nodes in the literature. We further argue, with the aid of a mathematical analysis, that our SPV method is generally effective for finding the symmetric nodes in real-world networks that typically do not have a dominant regular component. Our SPV-based framework, therefore, provides a physically intuitive and computationally efficient way to uncover, understand, and exploit symmetric structures in complex networks arising from real-world applications.

Smith–Treumann theory and the linkage principle

We apply Treumann’s “Smith theory for sheaves” in the context of the Iwahori–Whittaker model of the Satake category. We deduce two results in the representation theory of reductive algebraic groups over fields of positive characteristic: (a) a geometric proof of the linkage principle; (b) a character formula for tilting modules in terms of the ell -canonical basis, valid in all blocks and in all characteristics.

On Infinite Families of Narrow-Sense Antiprimitive BCH Codes Admitting 3-Transitive Automorphism Groups and Their Consequences

The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field $\mathrm{GF}(q)$ with length $q+1$, which are closely related to the action of the projective general linear group of degree two on the projective line. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some $q$-ary BCH codes with length $q+1$, and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of $\mathrm{PGL} (2, p^m)$-invariant codes over $\mathrm{GF} (p^h)$ is completed. As an application of this result, the $p$-ranks of all incidence structures invariant under the projective general linear group $\mathrm{ PGL }(2, p^m)$ are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a $3$-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms.

On simple modules with singular highest weights for so2l+1(K)

In this paper, we study formal characters of simple modules with singular highest weights over classical Lie algebras of type B over an algebraically closed field of characteristic p ≥ h, where h is the Coxeter number. Assume that the highest weights of these simple modules are restricted. We have given a description of their formal characters. In particular, we have obtained some new examples of simple Weyl modules. In the restricted region, the representation theory of algebraic groups and its Lie algebras are equivalent. Therefore, we can use the tools of the representation theory of semisimple and simply-connected algebraic groups in positive characteristic. To describe the formal characters of simple modules, we construct Jantzen filtrations of Weyl modules of the corresponding highest weights.

Coxeter combinatorics for sum formulas in the representation theory of algebraic groups

Let G G be a simple algebraic group over an algebraically closed field F \mathbb {F} of characteristic p ≥ h p \geq h , the Coxeter number of G G . We observe an easy ‘recursion formula’ for computing the Jantzen sum formula of a Weyl module with p p -regular highest weight. We also discuss a ‘duality formula’ that relates the Jantzen sum formula to Andersen’s sum formula for tilting filtrations and we give two different representation theoretic explanations of the recursion formula. As a corollary, we also obtain an upper bound on the length of the Jantzen filtration of a Weyl module with p p -regular highest weight in terms of the length of the Jantzen filtration of a Weyl module with highest weight in an adjacent alcove.

An algebraic groups perspective on Erdős–Ko–Rado

We give a proof of the Erdős–Ko–Rado Theorem using the Borel Fixed-Point Theorem from algebraic group theory. This perspective gives a strong analogy between the Erdős–Ko–Rado Theorem and (generalizations of) the Gerstenhaber Theorem on spaces of nilpotent matrices.

Multiplicity-free representations of algebraic groups II

We continue our work, started in [9], on the program of classifying triples (X,Y,V), where X,Y are simple algebraic groups over an algebraically closed field of characteristic zero with X<Y, and V is an irreducible module for Y such that the restriction V↓X is multiplicity-free. In this paper we handle the case where X is of type A, and is irreducibly embedded in Y of type B,C or D. It turns out that there are relatively few triples for X of arbitrary rank, but a number of interesting exceptional examples arise for small ranks.

On unitary representations of algebraic groups over local fields

Let G \mathbf {G} be an algebraic group over a local field k \mathbf {k} of characteristic zero. We show that the locally compact group G ( k ) \mathbf {G}(\mathbf {k}) consisting of the k \mathbf {k} -rational points of G \mathbf {G} is of type I. Moreover, we complete Lipsman’s characterization of the groups G \mathbf {G} for which every irreducible unitary representation of G ( k ) \mathbf {G}(\mathbf {k}) is a CCR representation and show at the same time that such groups belong to the family of trace class groups, recently studied by Deitmar and van Dijk.

Topological dynamics and NIP fields

We study definable topological dynamics of some algebraic group actions over an arbitrary NIP field K. We show that the Ellis group of the universal definable flow of SL2(K) is non-trivial if the multiplicative group of K is not type-definably connected, providing a way to find multiple counterexamples to the (model-theoretic) Ellis group conjecture, particularly in the case of dp-minimal fields. We also study some structure theory of algebraic groups over K with definable f-generics.