Quasirecognition by the set of element orders of the groups E 6( q ) and 2 E 6( q )

We prove that if L is one of the simple groups E ₆(q) and ² E ₆(q) and G is some finite group with the same spectrum as L, then the commutant of G/F(G) is isomorphic to L and the quotient G/G′ is a cyclic {2,3}-group. Original Russian Text Copyright ? 2007 Kondrat’ev A. S. The author was supported by the Russian Foundation for Basic Research (Grant 04-01-00463) and the RFBR-NSFC (Grant 05-01-39000). __________ Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 6, pp. 1250–1271, November–December, 2007.     

Irreducible characters with equal roots in the groups Sn and An

We show that treating of (non-trivial) pairs of irreducible characters of the group S_n sharing the same set of roots on one of the sets A_n and S_n \ A_n is divided into three parts. This, in particular, implies that any pair of such characters χ^α and χ^β (α and β are respective partitions of a number n) possesses the following property: lengths d(α) and d(β) of principal diagonals of Young diagrams for α and β differ by at most 1. Supported by RFBR grant No. 04-01-00463 and by RFBR-NSFC grant No. 05-01-39000. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 3–25, January–February, 2007.     

Expansion in SLd(Z/qZ), q arbitrary

Let S be a fixed finite symmetric subset of SL _d (Z), and assume that it generates a Zariski-dense subgroup G. We show that the Cayley graphs of π _q (G) with respect to the generating set π _q (S) form a family of expanders, where π _q is the projection map Z→Z/q Z.     

On the distribution of points in orbits of PGL(2,q) acting on GF(q)

We prove results on the distribution of points in an orbit of PGL(2,q) acting on an element of GF(qⁿ). These results support a conjecture of Klapper. More precisely, we show that the points in an orbit are uniformly distributed if n is small with respect to q.     

关于square-full数上的函数q_k~((e))(n)的均值估计

本文研究了指数k-free数的特征函数q_k~((e))(n)(k≥3)在square-full数集中的均值估计问题.利用黎曼Zeta函数的性质以及留数定理,获得了该均值的渐近公式,推广了q_k~((e))(n)在整数集中的均值估计相关结果.     

Quasirecognizability by the Set of Element Orders for Groups 3D4(q) and F4(q), for q Odd

It is proved that if L is one of the simple groups ³D₄(q) or F₄(q), where q is odd, and G is a finite group with the set of element orders as in L, then the derived subgroup of G/F(G) is isomorphic to L and the factor group G/G′ is a cyclic {2, 3}-group. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 517–539, September–October, 2005. Supported by RFBR grant No. 04-01-00463.     

Monomial realization of crystal graphs for Uq(An(1))

We give a new realization of crystal graphs for irreducible highest weight modules over U_q(A_n⁽¹⁾) in terms of the monomials introduced by H. Nakajima. We also discuss the natural connection between the monomial realization and other known realizations, path realization and Young wall realization.This research was supported by KOSEF Grant # 98-0701-01-5-L and BK21 Mathematical Sciences Division, Seoul National University.     

Computation of cohomology groups of the Lie algebras of type Bn and Cn

We adduce the results of the numerical experiment in calculating the spaces of local deformations of classical Lie algebras of type B _n and C _n in characteristic p = 2. Original Russian Text ? D.V. Reshetnikov, 2009, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2009, No. 8, pp. 71–72.     

On the k-error linear complexity of sequences with period 2pn over GF(q)

In this paper, we first optimize the structure of the Wei–Xiao–Chen algorithm for the linear complexity of sequences over GF(q) with period N =  2p ⁿ , where p and q are odd primes, and q is a primitive root modulo p ². The second, an union cost is proposed, so that an efficient algorithm for computing the k-error linear complexity of a sequence with period 2p ⁿ over GF(q) is derived, where p and q are odd primes, and q is a primitive root modulo p ². The third, we give a validity of the proposed algorithm, and also prove that there exists an error sequence e ^N , where the Hamming weight of e ^N is not greater than k, such that the linear complexity of (s + e) ^N reaches the k-error linear complexity c. We also present a numerical example to illustrate the algorithm. Finally, we present the minimum value k for which the k-error linear complexity is strictly less than the linear complexity.     

Lax Operator for the Quantised Orthosymplectic Superalgebra Uq[osp(m|n)]

Representations of quantum superalgebras provide a natural framework in which to model supersymmetric quantum systems. Each quantum superalgebra, belonging to the class of quasi-triangular Hopf superalgebras, contains a universal R-matrix which automatically satisfies the Yang–Baxter equation. Applying the vector representation π, which acts on the vector module V, to the left-hand side of a universal R-matrix gives a Lax operator. In this article a Lax operator is constructed for the quantised orthosymplectic superalgebras U _q [osp(m|n)] for all m > 2, n ≥ 0 where n is even. This can then be used to find a solution to the Yang–Baxter equation acting on V ⊗ V ⊗ W, where W is an arbitrary U _q [osp(m|n)] module. The case W = V is studied as an example. Presented by A. Verschoren.     

Quasi-permutation Representations of the Groups SU (3, q2) and PSU (3, q2)

A square matrix over the complex field with non-negative integral trace is called a quasi-permutation matrix. For a finite group G the minimal degree of a faithful permutation representation of G is denoted by p(G). The minimal degree of a faithful representation of G by quasi-permutation matrices over the rational and the complex numbers are denoted by q(G) and c(G) respectively. Finally r(G) denotes the minimal degree of a faithful rational valued complex character of G. In this paper p(G), q(G), c(G) and r(G) are calculated for the groups PSU (3, q²) and SU (3, q²).AMS Subject Classification (2000): 20C15     

The character table of a split extension of the Heisenberg group H 1( q ) by Sp (2, q ), q odd

In this paper we determine the full character table of a certain split extension of the Heisenberg group H ₁ by the odd-characteristic symplectic group Sp(2, q).      

Average Width and Optimal Interpolation of the Sobolev-Wiener Class Wpq(R) in the Metric Lq(R)

In this paper, we determine the exact value of average n − K width _n(W^r_pq(R), L_q(R)) of Sobolev-Wiener class W^r_pq(R) in the metric L_q(R) for 1 > q ≥ p > ∞ and get the value of _n(W^r_p(R), L_qp(R)) for the dual case. We also solve the optimal interpolation problems of W^r_pq(R) in the metric L_q(R) and W^r_p(R) in the metric L_qp(R) for 1 < q ≤ p < ∞.     

A generalization of adjoint crystals for the quantized affine algebras of type An(1) , Cn(1) and Dn+1(2)

We generalize Benkart-Frenkel-Kang-Lee’s adjoint crystals and describe their crystal structure for type A _n ⁽¹⁾, C _n ⁽¹⁾ and D _n+1⁽²⁾.     

Cyclotomic numbers and primitive idempotents in the ring GF(q)[x]/(x−1)

Let q be an odd prime power and p be an odd prime with gcd(p,q)=1. Let order of q modulo p be f, and q^f=1+pλ. Here expressions for all the primitive idempotents in the ring R_pⁿ=GF(q)[x]/(x^pn−1), for any positive integer n, are obtained in terms of cyclotomic numbers, provided p does not divide λ if n2. The dimension, generating polynomials and minimum distances of minimal cyclic codes of length pⁿ over GF(q) are also discussed.     

Almost simple groups with socle 3D4(q) act on finite linear spaces

After the classification of flag-transitive linear spaces, attention has now turned to line-transitive linear spaces. Such spaces are first divided into the point-imprimitive and the point-primitive, the first class is usually easy by the theorem of Delandtsheer and Doyen. The primitive ones are now subdivided, according to the O’Nan-Scotte theorem and some further work by Camina, into the socles which are an elementary abelian or non-abelian simple. In this paper, we consider the latter. Namely, T ≤ G ≤ Aut(T) and G acts line-transitively on finite linear spaces, where T is a non-abelian simple. We obtain some useful lemmas. In particular, we prove that when T is isomorphic to ³ D ₄(q), then T is line-transitive, where q is a power of the prime p.     

The young diagrams of a pair of irreducible characters of S n with the same zero set on S n ?

In studying the pairs of irreducible characters of the symmetric group S _n with the same zero set on A _n or S _n ∖ A _n (as well as the pairs of irreducible characters with the same zero set on the alternating group A _n ), the results are important on the connection between the Young diagrams of the characters of these pairs. We prove a theorem that considerably generalizes two previous results of frequent use in this direction. Original Russian Text Copyright ? 2008 Belonogov V. A. The author was supported by the Russian Foundation for Basic Research (Grant 07-01-00148) and the RFBR-NSFC (Grant 05-01-39000). __________ Ekaterinburg. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 49, No. 5, pp. 992–1006, September–October, 2008.     

Linear (q+1)-fold Blocking Sets in PG(2, q)

A (q+1)-fold blocking set of size (q+1)(q⁴+q²+1) in PG(2, q⁴) which is not the union of q+1 disjoint Baer subplanes, is constructed     

A conjecture of Abe-Iiyori and the Ree groups 2F4(q), II

In this paper the author has solved a problem of Abe and liyori for the finite simple groups ² F ₄(q) and ² F ₄(2)′.