Special Linear Group Of Degree Research Articles

Unit Group of the Group Algebra $\mathbb{F}_qGL(2,7)$

Unit Group of the Group Algebra $\mathbb{F}_qGL(2,7)$

The twisted Reidemeister torsion of an iterated torus knot

We calculate the twisted Reidemeister torsion of the complement of an iterated torus knot associated with a representation of its fundamental group to the complex special linear group of degree two. We also show that the twisted Reidemeister torsions associated with various representations appear in the asymptotic expansion of the colored Jones polynomial.

Magnetic curves in the real special linear group

We investigate contact magnetic curves in the real special linear group of degree 2. They are geodesics of the Hopf tubes over the projection curve. We prove that periodic contact magnetic curves in SL(2,R) can be quantized in the set of rational numbers. Finally, we study contact homogeneous magnetic trajectories in SL(2,R) and show that they project to horocycles in H2(-4).

Branching Rules for n-fold Covering Groups of SL2 over a Non-Archimedean Local Field

AbstractLet G be the n-fold covering group of the special linear group of degree two over a non-Archimedean local field. We determine the decomposition into irreducibles of the restriction of the principal series representations of G to a maximal compact subgroup. Moreover, we analyse those features that distinguish this decomposition from the linear case.

Centralizers in a group whose central factor is simple

In this paper, we find the number of the element centralizers of a finite group [Formula: see text] such that the central factor of [Formula: see text] is the projective special linear group of degree 2 or the Suzuki group. Our results generalize some main results of [Ashrafi and Taeri, On finite groups with a certain number of centralizers, J. Appl. Math. Comput. 17 (2005) 217–227; Schmidt, Zentralisatorverbände endlicher Gruppen, Rend. Sem. Mat. Univ. Padova 44 (1970) 97–131; Zarrin, On element centralizers in finite groups, Arch. Math. 93 (2009) 497–503]. Also, we give an application of these results.

A characterization of linear groups $$L_3(q)$$ L 3 ( q ) by their character degree graphs and orders

Let \(L_n(q)\) be the projective special linear group of degree n over a finite field of order q. In this paper, the projective special linear group \(L_3(q)\), where \(q\in \{5,7,8,9\}\) is studied using degree graph and its order.

A characterization of L 3(4) by its character degree graph and order.

Let G be a finite group. The character degree graph Gamma (G) of G is the graph whose vertices are the prime divisors of character degrees of G and two vertices p and q are joined by an edge if pq divides the character degree of G. Let L_n(q) be the projective special linear group of degree n over a finite field of order q. Khosravi et. al. have shown that the simple groups L_2(p^2), and L_2(p) where pin {7,8,11,13,17,19} are characterizable by the degree graphs and their orders. In this paper, we give a characterization of L_3(4) by using the character degree graph and its order.

On the existence of Engel pairs in certain linear groups

Let [Formula: see text] be a group and [Formula: see text]. The 2-tuple [Formula: see text] is said to be an [Formula: see text]-Engel pair, [Formula: see text], if [Formula: see text], [Formula: see text] and [Formula: see text]. Let SL[Formula: see text] be the special linear group of degree [Formula: see text] over the field [Formula: see text]. In this paper, we show that given any field [Formula: see text], there is a field extension [Formula: see text] of [Formula: see text] with [Formula: see text] such that SL[Formula: see text] has an [Formula: see text]-Engel pair for some integer [Formula: see text]. We will also show that SL[Formula: see text] has a [Formula: see text]-Engel pair if [Formula: see text] is a field of characteristic [Formula: see text].

The Colored Jones Polynomial, the Chern–Simons Invariant, and the Reidemeister Torsion of a Twice–Iterated Torus Knot

A generalization of the volume conjecture relates the asymptotic behavior of the colored Jones polynomial of a knot to the Chern–Simons invariant and the Reidemeister torsion of the knot complement associated with a representation of the fundamental group to the special linear group of degree two over complex numbers. If the knot is hyperbolic, the representation can be regarded as a deformation of the holonomy representation that determines the complete hyperbolic structure. In this article, we study a similar phenomenon when the knot is a twice-iterated torus knot. In this case, the asymptotic expansion of the colored Jones polynomial splits into sums, and each summand is related to the Chern–Simons invariant and the Reidemeister torsion associated with a representation.

On Linear Groups of Degree 2n Containing a Representation of the Special Linear Group of Degree n

Let n be an integer, n ≥ 2, and let a field P be a quadratic extension of an infinite field k. Regarding P as a k-vector space of dimension 2, we consider an n-dimensional P-vector space V as a 2n-dimensional k-vector space so the general linear group GL n (P) acting on V is embedded in the group GL 2n (k). Let a field K be an algebraic extension of k. In this article, we determine overgroups of the special linear group SL n (P) in the group GL 2n (K).

Some remarks on the ring of twisted tilting modules for algebraic groups

In a recent paper [S. Doty, A. Henke, Decomposition of tensor products of modular irreducibles for SL 2, Q. J. Math. 56 (2005) 189–207], Doty and Henke give a decomposition of the tensor product of two rational simple modules for the special linear group of degree 2 over an algebraically closed field of characteristic p > 0 . In performing this calculation it proved useful to know that the simple modules are twisted tensor products of tilting modules. It seems natural therefore to consider the ring of twisted tilting modules for a semisimple group G (a subring of the representation ring of G). However, we quickly specialize to the case in which G is the special linear group of degree 2. We show that (in this case) the ring is reduced and describe associated varieties. We give formulas from which one may determine the multiplicities of the indecomposable module summands of the tensor product of twisted tilting modules.

The cohomology of line bundles on the three-dimensional flag variety

We give a recursive description of the characters of the cohomology of the line bundles of the three-dimensional flag variety over an algebraically closed field of characteristic p. The recursion involves also certain rank two bundles and we calculate their cohomology at the same time. The method of proof is to adapt an expansion formula valid generally for homogeneous vector bundles on flag varieties G / B to the case in which G is the special linear group of degree 3. In order to carry this out it is necessary to calculate explicitly the module of invariants for the action of the first infinitesimal subgroup of the unipotent radical of a Borel subgroup of G on certain tilting modules.

A version of the volume conjecture

We propose a version of the volume conjecture that would relate a certain limit of the colored Jones polynomials of a knot to the volume function defined by a representation of the fundamental group of the knot complement to the special linear group of degree two over complex numbers. We also confirm the conjecture for the figure-eight knot and torus knots. This version is different from S. Gukov's because of a choice of polarization.

Symmetric generation and existence of [formula omitted], the automorphism group of the third Janko group

We make use of the primitive permutation action of degree 120 of the automorphism group of L 2 ( 16 ) , the special linear group of degree 2 over the field of order 16, to give an elementary definition of the automorphism group of the third Janko group J 3 and to prove its existence. This definition enables us to construct the 6156-point graph which is preserved by J 3 : 2 , to obtain the order of J 3 and to prove its simplicity. A presentation for J 3 : 2 follows from our definition, which also provides a concise notation for the elements of the group. We use this notation to give a representative of each of the conjugacy classes of J 3 : 2 .

A Conjugacy Theorem for Subgroups of SL n that Contain the Group of Diagonal Matrices

Let R be a commutative local ring. It is proved that if n ≥ 3 and the residue field of R contains at least 3n + 2 elements, then the subgroup of diagonal matrices in the special linear group of degree n over R is pronormal. For semilocal rings with the same restrictions on residue fields, this subgroup is paranormal. Bibliography: 13 titles.

Invariant Hyperfunction Solutions to Invariant Differential Equations on the Space of Real Symmetric Matrices

The real special linear group of degree n naturally acts on the vector space of n× n real symmetric matrices. How to determine invariant hyperfunction solutions of invariant linear differential equations with polynomial coefficients on the vector space of n× n real symmetric matrices is discussed in this paper. We prove that every invariant hyperfunction solution is expressed as a linear combination of Laurent expansion coefficients of the complex power of the determinant function with respect to the parameter of the power. Then the problem is reduced to the determination of Laurent expansion coefficients.

Computation of the Dimensions of Symmetry Classes of Tensors Associated with the Finite two Dimensional Projective Special Linear Group

The dimensions of the symmetry classes of tensors associated with the projective special linear group of degree 2 over a field with q elements, PSL2(q), are found. Of course we will assume PSL2(q) as a subgroup of the symmetric group Sq+1 because this group has a faithful action on the points of the underlying projective space. We also discuss the non-triviality of the symmetry classes of tensors associated with each irreducible character of PSL2(q).

On subgroup of the special linear group of degree 2 over an infinite field

We study the subgroups of over a field that comprise a conjugate in of the group of upper-unitriangular matrices of degree 2 over an infinite field such that is an algebraic extension.

On Weierstrass points of Riemann surfaces associated with $\Gamma(n,2n)$

Introduction. Let f be a congruence subgroup of SL(2, R), and let R(F) be the Riemann surface associated with F, i.e., R(F) is the canonical compactification of F\H, where H is the upper half plane of C. If the genus of R(F) is not less than two, then we can ask the following problems: Problem 1. Is R(F) hyperelliptic ? Problem 2. Are the cusps of R(F) Weierstrass points? Historically, Problems 1 and 2 are completely solved for F = F(n) by H. Petersson [8] and by B. Schoeneberg [9] respectively. In the case of F0(n), partial solutions are given by J. Lehener and M. Newmann [6] and A.O.L. Atkin [2]. The purpose of this note Is to answer both problems in the case of F = F(n, 2n) (as for the definition of I\n, 2ri), see Definition 1). Our results are the following: 1. R(I\n, 2n)) is non-hyperelliptic for any positive integer ≪i^4 (see Theorem 4). 2. Every cusp of R(F(n, 2n)) is a Weierstrass point for any even integer n^4. But there is an example, where the opposite situation may occur if n is odd (see Theorem 6 and Remark 1). Notation. SL{2, R) (resp. SL(2, Z)) is the special linear group of degree two over the real number field R (resp. the rational integer ring Z), and PSL(2, R) is the projective special linear group of degree two over R. When J is a subgroup of SL(2, R), the image of J under the canonical homomorphism SL(2, R)―>PSL (2, R) is denoted by A. H* means the disjoint union of the upper half plane H, the rational numbers Q and {oo}.

ON THE STRUCTURE OF THE SPECIAL LINEAR GROUP OVER POLYNOMIAL RINGS

It is proved that the special linear group of degree not less than three over the polynomial ring over a field is generated by the elementary matrices. Other results are obtained that relate to the structure of the special linear group and stabilization of the general linear group over arbitrary polynomial and Laurent rings.Bibliography: 9 titles.