Subgroup Of General Linear Group Research Articles

Regularity and stable ranges of FI-modules

We give refined bounds for the regularity of FI-modules and the stable ranges of FI-modules for various forms of their stabilization studied in the representation stability literature. We show that our bounds are sharp in several cases. We apply these to get explicit stable ranges for diagonal coinvariant algebras, and improve those for ordered configuration spaces of manifolds and congruence subgroups of general linear groups.

Second Subgroup of General Linear Group in Dimention 2

In many practical situations we know that a primitive group contains a given permutation and we want to know which group it can be; in some other practical situations we know the group and would like to know if it contains a permutation of some given type. For example, a group G ≤ Sn is said to be non-synchronizing if it is contained in the automorphism group of a non-trivial primitive graph with complete core, that is, with clique number equal to chromatic number. When trying to check if some group is synchronizing, typically, we have only partial information about the graph but enough to say that it has an automorphism of some type, and the goal would be to have in hand a classification of the primitive groups containing permutations of that type. As an illustration of this, the key ingredient in some of the results in [2] was the observation that the primitive graph understudy has a 2-cycle automorphism and hence the automorphism group of the graph is the symmetric group. For many more examples of the importance of knowing the groups that contain permutations of a given type. This type of investigation is certainly very natural since it appears on the eve of group theory, with Jordan, Burnside, Marggraff, but the difficulty of the problem is well illustrated by the very slow progress throughout the twentieth century. Given the new tools available (chiefly the classification of finite simple groups), the topic seems to have new momentum. Let Sn denote the symmetric group on n points; a permutation g∈Sn is said to be imprimitive if there exists an imprimitive group contain in gg. An imprimitive group is said to beminimally imprimitiveifit contains no transitive proper subgroup. An imprimitive group G≤Sn is said to be maximally imprimitive if for all g Sn\G, the group〈g, G 〉 is primitive. The next result, whose proof isstraightforward, provides some alternative characterizations of imprimitive permutations. Now in this paper we discuse Presentation for imprimitive second Subgroup of general linear group in dimention 2 over the field of pk-elements .

Alternating sign matrices of finite multiplicative order

We investigate alternating sign matrices that are not permutation matrices, but have finite order in a general linear group. We classify all such examples of the form P+T, where P is a permutation matrix and T has four non-zero entries, forming a square with entries 1 and −1 in each row and column. We show that the multiplicative orders of these matrices do not always coincide with those of permutation matrices of the same size. We pose the problem of identifying finite subgroups of general linear groups that are generated by alternating sign matrices.

On Finite Subgroups in the General Linear Groups over an Algebraic Number Field

As is well-known, there are only finitely many isomorphic classes of finite subgroups in a given general linear group over the field of rational numbers. This result can be generalized to any algebraic number field. While the case of field of rational numbers is relatively well-studied, we still do not know much for general algebraic number field cases. In this article, we discuss the finiteness of isomorphic classes of finite subgroups of general linear groups over an algebraic number field. We give a method to calculate a multiplicative bound for the orders of finite subgroups and to classify finite cyclic subgroups.

Complex Tori, Theta Groups and Their Jordan Properties

We prove that an analog of Jordan’s theorem on finite subgroups of general linear groups does not hold for the group of bimeromorphic automorphisms of a product of the complex projective line and a complex torus of positive algebraic dimension.

Product of symplectic groups and its cyclic orbit code

Constant dimension subspace codes are subsets of the finite Grassmann Variety. Orbit codes are constant dimension subspace codes that arise as the orbit of subgroup of general linear group acting on subspaces in an ambient space. In particular, orbit codes of symplectic subgroup of the general linear group have been investigated recently. In this paper, we determine product of symplectic groups and its orbit code, and decoding algorithm of this code is considered.

Linear stable range for homology of congruence subgroups via FI-modules

We answer positively a question of Church, Miller, Nagpal and Reinhold on existence of a linear bound on the presentation degree of the homology of a complex of FI-modules. This implies a linear stable range for the homology of congruence subgroups of general linear groups.

Linear and quadratic ranges in representation stability

We prove two general results concerning spectral sequences of FI-modules. These results can be used to significantly improve stable ranges in a large portion of the stability theorems for FI-modules currently in the literature. We work this out in detail for the cohomology of configuration spaces where we prove a linear stable range and the homology of congruence subgroups of general linear groups where we prove a quadratic stable range. Previously, the best stable ranges known in these examples were exponential. Up to an additive constant, our work on congruence subgroups verifies a conjecture of Djament.

Menon-type identities derived from actions of subgroups of general linear groups

In this article, we consider the actions of subgroups of the general linear group GLr(Zn) on Znr, including groups of upper triangular matrices in GLr(Zn), unipotent groups, Heisenberg groups and extended Heisenberg groups. By applying the Cauchy–Frobenius–Burnside lemma, we obtain several generalizations of the well known Menon's identity.

Cryptanalysis of cryptosystems based on general linear group

Advances in quantum computers threaten to break public key cryptosystems such as RSA, ECC, and EIGamal on the hardness of factoring or taking a discrete logarithm, while no quantum algorithms are found to solve certain mathematical problems on non-commutative algebraic structures until now. In this background, Majid Khan et al. proposed two novel public-key encryption schemes based on large abelian subgroup of general linear group over a residue ring. In this paper we show that the two schemes are not secure. We present that they are vulnerable to a structural attack and that, it only requires polynomial time complexity to retrieve the message from associated public keys respectively. Then we conduct a detailed analysis on attack methods and show corresponding algorithmic description and efficiency analysis respectively. After that, we propose an improvement assisted to enhance Majid Khan's scheme. In addition, we discuss possible lines of future work.

On automorphic L-functions in positive characteristic

We give a proof of the existence of Asai, exterior square, and symmetric square local $L$-functions, $\gamma$-factors and root numbers in characteristic $p$, including the case of $p = 2$. Our study is made possible by developing the Langlands-Shahidi method over a global function field in the case of a Siegel Levi subgroup of a split classical group or a quasi-split unitary group. The resulting automorphic $L$-functions are shown to satisfy a rationality property and a functional equation. A uniqueness result of the author and G. Henniart allows us to show that the definitions provided in this article are in accordance with the local Langlands conjecture for ${\rm GL}_n$. Furthermore, in order to be self contained, we include a treatise of $L$-functions arising from maximal Levi subgroups of general linear groups.

A Novel Cryptosystem Based on General Linear Group

In this article, we have developed a novel public-key cryptosystem that uses large abelian subgroup of general linear group over residue ring. The merit of this proposed public key cryptosystem is that we can select session key in abelian subgroup of general linear group which reduces exponentiations and performed encryption effectively in a simple way. Our algorithm doesn't use matrix modular exponentiation which leads us to problem in implementing. The aim of this article is to decrease the number of these exponentiations and consequently to accelerate the execution of encryption algorithm. A discussion about the security of built modifications made in the article shows that the level of security is high enough for an appropriate choice of parameters of the cryptosystems.

JORDAN GROUPS AND ALGEBRAIC SURFACES

We prove that an analogue of Jordan’s theorem on finite subgroups of general linear groups holds for the groups of biregular automorphisms of algebraic surfaces. This gives a positive answer to a question of Vladimir L. Popov.

JORDAN GROUPS AND ELLIPTIC RULED SURFACES

We prove that an analogue of Jordan’s theorem on finite subgroups of general linear groups holds for the groups of biregular automorphisms of elliptic ruled surfaces. This gives a positive answer to a question of Vladimir L. Popov.

Theta Groups and Products of Abelian and Rational Varieties

AbstractWe prove that an analogue of Jordan's theorem on finite subgroups of general linear groups does not hold for the groups of birational automorphisms of products of an elliptic curve and the projective line. This gives a negative answer to a question posed by Vladimir L. Popov.

Conjugacy classes in parabolic subgroups of general linear groups

We prove a formula connecting the number of unipotent conjugacy classes in a maximal parabolic subgroup of a finite general linear group with the numbers of unipotent conjugacy classes in various parabolic subgroups in smaller dimensions. We generalise this formula and deduce a number of corollaries; in particular, we express the number of conjugacy classes of unitriangular matrices over a finite field in terms of the numbers of unipotent conjugacy classes in maximal parabolic subgroups over the same field. We show how the numbers of unipotent conjugacy classes in parabolic subgroups of small dimensions may be calculated.

Prehomogeneous spaces for Borel subgroups of general linear groups

Let k be an algebraically closed field. Let B be the Borel subgroup of GLn(k) consisting of nonsingular upper triangular matrices. Let b = Lie B be the Lie algebra of upper triangular n × n matrices and u the Lie subalgebra of b consisting of strictly upper triangular matrices. We classify all Lie ideals n of b, satisfying u' ⫅ n ⫅ u, such that B acts (by conjugation) on n with a dense orbit. Further, in case B does not act with a dense orbit, we give the minimal codimension of a B-orbit in n. This can be viewed as a first step towards the difficult open problem of classifying of all ideals n ⫅ u such that B acts on n with a dense orbit. The proofs of our main results require a translation into the representation theory of a certain quasi-hereditary algebra At,1. In this setting we find the minimal dimension of Ext1At,1(M,M) for a δ-good At,1-module of certain fixed δ-dimension vectors.

Representations of Parabolic and Borel Subgroups

We demonstrate a relationships between the representation theory of Borel subgroups and parabolic subgroups of general linear groups. In particular, we show that the representations of Borel subgroups could be computed from representations of certain maximal parabolic subgroups.

Conjugacy Classes in Maximal Parabolic Subgroups of General Linear Groups

We compute conjugacy classes in maximal parabolic subgroups of the general linear group. This computation proceeds by reducing to a “matrix problem.” Such problems involve finding normal forms for matrices under a specified set of row and column operations. We solve the relevant matrix problem in small dimensional cases. This gives us all conjugacy classes in maximal parabolic subgroups over a perfect field when one of the two blocks has dimension less than 6. In particular, this includes every maximal parabolic subgroup of GLn(k) for n<12 and k a perfect field. If our field is finite of size q, we also show that the number of conjugacy classes, and so the number of characters, of these groups is a polynomial in q with integral coefficients.

Modular Invariants of Parabolic Subgroups of General Linear Groups

Let p be a prime number and let A be an elementary abelian p-group of rank m. The purpose of this paper is to determine a full system for the invariants of parabolic subgroups of the general linear group GL(m,Fp) in H*(A,Fp). A relation between these invariants and Dickson ones is also obtained.