Sets of Elements that Pairwise Generate a Matrix Ring

In this article, we investigate the relationship between the minimum number of proper subgroups of GL(n, q) whose union is the whole GL(n, q) and the maximum number of elements that pairwise generate GL(n, q). We show that the minimum number of proper subrings of M _n (q) whose union is the whole M _n (q) is exactly the maximum number of elements that pairwise generate M _n (q).     

Parabolic conjugacy in general linear groups

Let q be a power of a prime and n a positive integer. Let P(q) be a parabolic subgroup of the finite general linear group GL _n (q). We show that the number of P(q)-conjugacy classes in GL _n (q) is, as a function of q, a polynomial in q with integer coefficients. This answers a question of Alperin in (Commun. Algebra 34(3): 889–891, 2006)     

Fischer Matrices of the Affine Groups

Let GL_n(q) be the general linear group and let H_n ; V_n(q) · GL_n(q) denote the affine group of V_n(q). In [1] and [4], we determined Fischer matrices for the conjugacy classes of GL_n(q) where n = 2, 3, 4 and we obtained the number of conjugacy classes and irreducible characters of H₂, H₃, and H₄. In this paper, we find the Fischer matrices of the affine group H_n for arbitrary n.AMS Subject Classification Primary 20C15 Secondary 20C33     

Nilpotent Injectors in General Linear Groups

In a recent paper, A. Bialostocki (Israel J. Math.41 (1982), 261-273) has defined a nilpotent injector in an arbitrary finite group G to be a maximal nilpotent subgroup of G, containing a subgroup H of G of maximal order satisfying class (H) ≤2. In the present paper, the author determines the nilpotent injectors of GL(n, q) and shows that they form a unique conjugacy class of subgroups of GL(n, q). It is also proved that if n ≠ 2 or n = 2 and q ≠ 9 is not a Fermat prime >3, then the nilpotent injectors of GL(n, q) are the nilpotent subgroups of maximal order.     

Order of elements in the groups related to the general linear group

For a natural number n and a prime power q the general, special, projective general and projective special linear groups are denoted by GL_n(q), SL_n(q), PGL_n(q) and PSL_n(q), respectively. Using conjugacy classes of elements in GL_n(q) in terms of irreducible polynomials over the finite field GF(q) we demonstrate how the set of order elements in GL_n(q) can be obtained. This will help to find the order of elements in the groups SL_n(q), PGL_n(q) and PSL_n(q). We also show an upper bound for the order of elements in SL_n(q).     

Representations of the Quantum Algebra Uq(un, 1)

The main aim of the paper is to study infinite-dimensional representations of the real form U _q (u _{n, 1}) of the quantized universal enveloping algebra U _q (gl _{n + 1}). We investigate the principal series of representations of U _q (u _{n, 1}) and calculate the intertwining operators for pairs of these representations. Some of the principal series representations are reducible. The structure of these representations is determined. Then we classify irreducible representations of U _q (u _{n, 1}) obtained from irreducible and reducible principal series representations. All *-representations in this set of irreducible representations are separated. Unlike the classical case, the algebra U _q (u _{n, 1}) has finite-dimensional irreducible *-representations.     

Group-theoretical generalization of necklace polynomials

Let G be a group, U a subgroup of G of finite index, X a finite alphabet and q an indeterminate. In this paper, we study symmetric polynomials M _G (X,U) and M_G^q(X,U)M_{G}^{q}(X,U) which were introduced as a group-theoretical generalization of necklace polynomials. Main results are to generalize identities satisfied by necklace polynomials due to Metropolis and Rota in a bijective way, and to express M_G^q(X,U)M_{G}^{q}(X,U) in terms of M _G (X,V)’s, where [V] ranges over a set of conjugacy classes of subgroups to which U is subconjugate. As a byproduct, we provide the explicit form of the GL _m (ℂ)-module whose character is M_\mathbbZ^q(X,n\mathbbZ)M_{\mathbb{Z}}^{q}(X,n\mathbb{Z}), where m is the cardinality of X.     

Coinvariants for a coadjoint action of quantum matrices

Let K be a (algebraically closed ) field. A morphism A ⟼ g ⁻¹ Ag, where A ∈ M(n) and g ∈ GL(n), defines an action of a general linear group GL(n) on an n × n-matrix space M(n), referred to as an adjoint action. In correspondence with the adjoint action is the coaction α: K[M(n)] → K[M(n)] ⊗ K[GL(n)] of a Hopf algebra K[GL(n)] on a coordinate algebra K[M(n)] of an n × n-matrix space, dual to the conjugation morphism. Such is called an adjoint coaction. We give coinvariants of an adjoint coaction for the case where K is a field of arbitrary characteristic and one of the following conditions is satisfied: (1) q is not a root of unity; (2) char K = 0 and q = ±1; (3) q is a primitive root of unity of odd degree. Also it is shown that under the conditions specified, the category of rational GL _q × GL _q -modules is a highest weight category.     

Linear groups over a skew field of quaternions containing the group T_{n}(A, \Phi) over a subsfield

We study the subgroups of GL_n(D) (n \geqq 3) GL_{n}(D) (n \geqq 3) over a skew field of quaternions D that comprise the subgroup of the unitary group U_n(A, F) U_{n}(A, \Phi) over a subsfield A \subseteqq D A \subseteqq D generated by all transvections in U_n(A, F) U_{n}(A, \Phi) .     

Representations of finite special linear groups in non-defining characteristic

We classify irreducible modules over the finite special linear group SLn(q) in the non-defining characteristic ?, describe restrictions of irreducible modules from GLn(q) to SLn(q), classify complex irreducible characters of SLn(q) irreducible modulo l, and discuss unitriangularity of the l-decomposition matrix for SLn(q).     

Braid group action via GL n (q) and U n (q), and Galois realizations

We determine the braid group action on generating systems of a group that is the semi-direct product of a finite vector space with a group of scalars. This leads to Galois realizations of certain groups GL _n (q) and PU _n (q). Dedicated to Prof. J. G. Thompson Partially supported by an NSA grant. This work was done while the author was a fellow of the Institute for Advanced Studies in Jerusalem.     

Exchange Rings Generated by Their Units

Let R be an exchange ring with primitive factors artinian. We prove that there exists a u∈U（R） such that 1R ± u ∈ U（R）, if and only if for any a ∈ R, there exists a u ∈ U（R） such that a ± u∈ U（R）. Phrthermore, we prove that, for any A ∈ Mn（R）（n ≥ 2）, there exists a U ∈ GLn（R） such that A ± U ∈ GLn（R）.     

Association Schemes of Quadratic Forms and Symmetric Bilinear Forms

Let X _n and Y _n be the sets of quadratic forms and symmetric bilinear forms on an n-dimensional vector space V over , respectively. The orbits of GL _n( ) on X _n × X _n define an association scheme Qua(n, q). The orbits of GL _n( ) on Y _n × Y _n also define an association scheme Sym(n, q). Our main results are: Qua(n, q) and Sym(n, q) are formally dual. When q is odd, Qua(n, q) and Sym(n, q) are isomorphic; Qua(n, q) and Sym(n, q) are primitive and self-dual. Next we assume that q is even. Qua(n, q) is imprimitive; when (n, q) (2,2), all subschemes of Qua(n, q) are trivial, i.e., of class one, and the quotient scheme is isomorphic to Alt(n, q), the association scheme of alternating forms on V. The dual statements hold for Sym(n, q).     

Mutually orthogonal Latin squares based on general linear groups

Given a finite group G, how many squares are possible in a set of mutually orthogonal Latin squares based on G? This is a question that has been answered for a few classes of groups only, and for no nonsoluble group. For a nonsoluble group G, we know that there exists a pair of orthogonal Latin squares based on G. We can improve on this lower bound when G is one of GL(2, q) or SL(2, q), q a power of 2, q ≠ 2, or is obtained from these groups using quotient group constructions. For nonsoluble groups, that is the extent of our knowledge. We will extend these results by deriving new lower bounds for the number of squares in a set of mutually orthogonal Latin squares based on the group GL(n, q), q a power of 2, q ≠ 2.     

On a continued fraction of order 12

We present some new relations between a continued fraction U(q) of order 12 (established by M. S. M. Naika et al.) and U(q ⁿ ) for n = 7, 9, 11, 13:     

The maximum element order in the groups related to the linear groups which is a multiple of the defining characteristic

Let n be a natural number and q be the power of a prime p. The general, special and projective special linear groups are denoted by GL_n(q), SL_n(q) and PSL_n(q), respectively. In this paper we find the maximum order of an element of the above groups which is a multiple of p.     

Actions of GLq (2,C) on C(1,3) and its four dimensional representations

A complete classification is given of all inner actions on the Clifford algebra C(l,3) defined by representations of the quantum group GL_q (2,C)q^m ≠1, which are not reduced to representations of two commuting “q-spinors”. As a consequence of this classification it is shown that the space of invariants of every GL_q (2,C)-action of this type, which is not an action of SL_q (2,C), is generatedby 1 and the value of the quantum determinant for the given representation.     

The number of connected sparsely edged graphs. II. Smooth graphs and blocks

A smooth graph is a connected graph without endpoints; f(n, q) is the number of connected graphs, v(n, q) is the number of smooth graphs, and u(n, q) is the number of blocks on n labeled points and q edges: W_k, V_k, and U_k are the exponential generating functions of f(n, n + k), v(n, n + k), and u(n, n + k), respectively. For any k ? 1, our reduction method shows that V_k can be deduced at once from W_k, which was found for successive k by the computer method described in our previous paper. Again the reduction method shows that U_k must be a sum of powers (mostly negative) of 1 - X and, given this information, we develop a recurrence method well suited to calculate U_k for successive k. Exact formulas for v(n, n + k) and u(n, n + k) for general n follow at once.     

Degenerate principal series and local theta correspondence II

Following our previous paper [LZ] which deals with the groupU(n, n), we study the structure of certain Howe quotients Ω ^p,q and Ω ^p,q (1) which are natural Sp(2n,R) modules arising from the Oscillator representation associated with the dual pair (O(p, q), Sp(2n,R)), by embedding them into the degenerate principal series representations of Sp(2n,R) studied in [L2].     

q-Deformed Euclidean algebras and their representations

A new q-deformed Euclidean algebra U_q (iso _n ), based on a definition of the algebra U_q (so _n ) different from the Drinfeld-Jimbo definition, is given. Infinite-dimensional representations T_a of this algebra, characterized by one complex number, is described. Explicit formulas for operators of these representations in an orthonormal basis are derived. The spectrum of the operator T_a(I_n) corresponding to a q-analogue of the infinitesimal operator of shifts along the n-th axis is given. Contrary to the case of the classical Euclidean algebraiso _n, this spectrum is discrete and the spectrum points have one point of accumulation.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 3, pp. 467–475, June, 1995.