Linear Group Research Articles

Super Multiset RSK and a Mixed Multiset Partition Algebra

Through dualities on representations on tensor powers and symmetric powers respectively, the partition algebra and multiset partition algebra have been used to study long-standing questions in the representation theory of the symmetric group. In this paper we extend this story to exterior powers, introducing the mixed multiset partition algebra as well as a generalization of the Robinson-Schensted-Knuth algorithm to two-row arrays of multisets with elements from two alphabets. From this algorithm, we obtain enumerative results that reflect representation-theoretic decompositions of this algebra. Furthermore, we use the generalized RSK algorithm to describe the decomposition of a polynomial ring in sets of commuting and anti-commuting variables as a module over both the general linear group and the symmetric group.

The multiplicative Lie algebra on general linear groups

Abstract The main aim of this paper is to study an obvious linear representation of a multiplicative Lie algebra. Also, we find some criteria to determine all possible multiplicative Lie algebra structures on a general linear group and we show that the general linear group on a finite field is a Lie simple group.

From HgGa2(SeO3)4 to Hg2Ga(SeO3)2F: The first HgI-based selenite birefringent crystal triggered by linear groups and fluoride ions

From HgGa2(SeO3)4 to Hg2Ga(SeO3)2F: The first HgI-based selenite birefringent crystal triggered by linear groups and fluoride ions

Topology and Automorphism Structures in General Linear Group

The general linear group, as a significant topic in algebra and topology, has a wide-ranging research background and application value. With the continuous advancement of mathematical research, the combination of topological structures and automorphism structures has become a key entry point for exploring the properties of the general linear group. This paper investigates the topological properties of general linear group GL (n, R) , specifically focusing on compactness, connectedness and the fundamental group, and the automorphism. The first part of the study is dedicated to a detailed analysis of these properties, providing new insights into the topological structural characteristics of GL (n, R) . This paper scrutinized the homotopy type within it, giving proof to the fundamental group of GL (n, R) , which is showed to be isomorphism to a trivial group. In the second half, This paper introduce and examine the function φ (G A GL n G A G ) = { ( , )| • } ∈ =  , which is used to identify the automorphism group of a dense subgroup G of Rn . Specifically, the automorphism group of Qn is investigated. This paper show that the automorphism group of it is isomorphic to the subgroup GL (n, Q) of general linear group GL (n, R) . Through this function, this paper offer an innovative approach to understanding the automorphisms within general linear groups, revealing the deep connections between algebraic and topological aspects of these groups.

On Kursov's theorem for matrices over division rings

Let D be a division ring with center F and multiplicative group D×, where each element of the commutator subgroup of D× can be expressed as a product of at most s commutators. A known theorem of Kursov states that if D is finite-dimensional over F, then every element of the commutator subgroup of the general linear group over D can be expressed as a product of at most s+1 commutators. We show that this result remains valid when F has a sufficiently large number of elements, without requiring D to be finite-dimensional. Our approach not only improves upon recent results on matrix decompositions over division rings but also provides a look at the Engel word map for matrices over arbitrary algebras.

Extended Special Linear group ESL2(F) and matrix equations in SL2(F), SL2(Z) and GL2(Fp)

The problem of roots existence for different classes of matrix such as simple and semisimple matrices from SL2(F), SL2(Z) and GL2(F) are solved. For this purpose, we first introduced the concept of an extended special linear group ESL2(F), which is generalisation of the matrix group SL2(F), where F is arbitrary perfect field. The group of unimodular matrices and extended symplectic group ESp2(R) are generalised by us, their structures are found. Our criterion oriented on a general class of matrix depending of the form of minimal and characteristic polynomials, moreover a proposed criterion holds in GL2(F), where F is an arbitrary field. The method of matrix factorisation is outlined. We show that ESL2(F) is a set of all square matrix roots from SL2(F) except of that established in our root existence criterion.

Isometries of Lipschitz‐free Banach spaces

AbstractWe describe surjective linear isometries and linear isometry groups of a large class of Lipschitz‐free spaces that includes, for example, Lipschitz‐free spaces over any graph. We define the notion of a Lipschitz‐free rigid metric space whose Lipschitz‐free space only admits surjective linear isometries coming from surjective dilations (i.e., rescaled isometries) of the metric space itself. We show that this class of metric spaces is surprisingly rich and contains all 3‐connected graphs as well as geometric examples such as nonabelian Carnot groups with horizontally strictly convex norms. We prove that every metric space isometrically embeds into a Lipschitz‐free rigid space that has only three more points.

On a polynomial bound for the orbital diameter of primitive affine groups

Let VG be a finite primitive affine permutation group, where V is a vector space of dimension d over the prime field F p and G is an irreducible linear group on V. We prove that if p divides |G|, then the diameters of all nondiagonal orbital graphs of VG are at most 9 d 3 . This improves an earlier exponential bound by A. Maróti and the author.

On eventually injective endomorphisms

We describe a property of linear groups that strengthens the Hopf property of finitely generated residually finite groups.

Amenability of quadratic automaton groups

We give lower bounds for the electrical resistance between vertices in the Schreier graphs of the action of the linear (degree 1) and quadratic (degree 2) mother groups on the orbit of the 0-ray. These bounds, combined with results of Juschenko et al. (2016), show that every quadratic activity automaton group is amenable. The resistance bounds use an apparently new “weighted” version of the Nash-Williams criterion which may be of independent interest.

A local-global principle for similarities over function fields of p-adic curves

Let p∈N be a prime with p≠2, and let K be a p-adic field. Let F be the function field of a curve over K. Let ΩF be the set of all divisorial discrete valuations of F. In this paper, we ask whether the Hasse principle holds for semisimple adjoint linear algebraic groups over F. We give a positive answer to this question for a class of adjoint classical groups.

Depth zero supercuspidal representations of classical groups into l-packets: The typically almost symmetric case

We classify what we call “typically almost symmetric” depth zero supercuspidal representations of a classical group over a local field of odd residual characteristic into L-packets. Our main results resolve an ambiguity in the paper of Lust-Stevens [24] in this case, where they could only classify these representations in two or four, if not one, L-packets. By assuming the expected numbers of supercuspidal representations in the L-packets, we employ only simple properties of the representations to prove the main results. In particular, we do not require any deep calculations of character values. With the same method, we also compute the parity of a (conjugate-)self-dual depth zero supercuspidal representation of a general linear group.

Sex matters: Differences in prodromes, clinical and neuropsychological features in individuals with a first episode mania or psychosis

ObjectiveThis study was aimed at identifying sex differences in patients presenting a first episode mania (FEM) or psychosis (FEP) to help shaping early treatment strategies focused on sex differences. MethodsPatients with a FEM or FEP underwent a clinical, neuropsychological (neurocognitive functions and emotional intelligence) and functional assessment. Performance on those variables was compared between groups through general linear model, with sex and group (FEM vs FEP) as main effects and group by sex interactions. ResultsThe total sample included 113 patients: FEM = 72 (45.83 % females) and FEP = 41 (46.34 % females). There were significant main effects for group (not for sex) for most of the clinical features (depressive, negative and positive symptoms) and psychosocial functioning (χ2 = 8.815, p = 0.003). As for neuropsychological performance, there were significant main effects for sex and group. Females performed better than males in verbal memory (χ2 = 9.038, p = 0.003) and obtained a higher emotional intelligence quotient (χ2 = 13.20, p < 0.001). On the contrary, males obtained better results in working memory (χ2 = 7.627, p = 0.006). FEP patients significantly underperformed FEM patients in most cognitive domains. There were significant group by sex interactions for few neuropsychological variables, namely processing speed (χ2 = 4.559, p = 0.033) and verbal fluency (χ2 = 8.913, p = 0.003). LimitationsDifferences between sexes were evaluated, but the influence of gender was not considered. Retrospective evaluation of prodromes and substance use. No healthy control group comparator. ConclusionThe main finding is the presence of significant sex effect and group by sex interaction on specific neurocognitive cognition and emotional intelligence measures. Tailored sex-based early treatment strategies might be implemented.

Standard monomials and invariant theory for arc spaces III: Special linear group

This is the third in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field K, we prove the arc space analogue of the first and second fundamental theorems of invariant theory for the special linear group. This is more subtle than the results for the general linear and symplectic groups obtained in the first two papers because the arc space of the corresponding affine quotients can be nonreduced.

Almost cyclic elements in primitive finite linear groups

Almost cyclic elements in primitive finite linear groups

Secret sharing in a special linear group

Objectives. The problem of developing the mathematical foundations of modular secret sharing in a special linear group over the ring of integers is being solved.The relevance of the problem is reduced to the fact that a large number of requirements are imposed on secret sharing schemes. These include the ideality of the scheme, the possibility of verification, changing the threshold without the participation of the dealer, the implementation of a non-threshold access structure and some others. Every secret sharing scheme developed to date does not fully satisfy all these requirements. It only has a certain configuration of these properties. The development of a scheme on a new mathematical basis is intended to expand the list of these configurations, which creates more opportunities for the user in choosing the optimal option.Methods. Group theory, modular arithmetic and theory of secret sharing schemes are used.Results. A fundamental domain with respect to the action of the main congruence subgroup by right shifts in the special linear group of second-order matrices over the ring of integers is constructed. On this basis, methods for modular secret sharing and its threshold restoration are proposed.Conclusion. A rigorous mathematical justification is given for the correctness of the algorithms for generating partial secrets and restoring the main secret in the special linear group over the ring of integers. These results will be used to study the configuration of secret sharing properties in this group.

Linkage and translation for tensor products of representations of simple algebraic groups and quantum groups

AbstractLet be either a simple linear algebraic group over an algebraically closed field of characteristic or a quantum group at an ‐th root of unity. We define a tensor ideal of singular ‐modules in the category of finite‐dimensional ‐modules and study the associated quotient category , called the regular quotient. For , the Coxeter number of , we establish a ‘linkage principle’ and a ‘translation principle’ for tensor products: Let be the essential image in of the principal block of . We first show that is closed under tensor products in . Then we prove that the monoidal structure of is governed to a large extent by the monoidal structure of . These results can be combined to give an external tensor product decomposition , where denotes the Verlinde category of .

Metric ultraproducts of groups — Simplicity, perfectness and torsion

We characterise the simplicity of metric ultraproducts of a family of metric groups. We also present several new examples of simple groups, such as metric ultraproducts of finite and infinite symmetric groups, linear groups, and interval exchange transformation groups. Using similar methods, we also examine concepts such as genericity, perfectness, and torsion.

On Recognition of Low-Dimensional Linear and Unitary Groups by Spectrum

On Recognition of Low-Dimensional Linear and Unitary Groups by Spectrum

ENUMERATION OF GROUPS IN SOME SPECIAL VARIETIES OF A-GROUPS

Abstract We find an upper bound for the number of groups of order n up to isomorphism in the variety ${\mathfrak {S}}={\mathfrak {A}_p}{\mathfrak {A}_q}{\mathfrak {A}_r}$ , where p, q and r are distinct primes. We also find a bound on the orders and on the number of conjugacy classes of subgroups that are maximal amongst the subgroups of the general linear group that are also in the variety $\mathfrak {A}_q\mathfrak {A}_r$ .