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.     

The invariant field of the adjoint action of the unitriangular group in the nilradical of a parabolic subalgebra

In the present paper the invariant field of the adjoint action of the unitriangular group in the nilradical of any parabolic subalgebra is described. Bibliography: 7 titles.     

On difference between the spectra of the simple groups Bn(q) and Cn(q)

We prove that the nonisomorphic simple groups B _n (q) and C _n (q) have different sets of element orders. Original Russian Text Copyright ? 2007 Grechkoseeva M. A. __________ Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 1, pp. 89–92, January–February, 2007.     

Automorphisms of elementary adjoint Chevalley groups of types Al, Dl, and El over local rings with 1/2

It is proved that every automorphism of an elementary adjoint Chevalley group of type A _l , D _l , or E _l over a local commutative ring with 1/2 is a composition of a ring automorphism and conjugation by some matrix from the normalizer of that Chevalley group in GL(V) (V is an adjoint representation space).     

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.     

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⁾.     

A Korovkin-Type Result in Ck an Application to the Mn Operators

In this work we present a result about the approximation of the k-th derivative of a function by means of a linear operator under assumptions related to shape preserving properties. As a consequence we deduce new results about the Meyer-König and Zeller operators.     

Inner Functions and Cyclic Composition Operators on H(Bn)

The present paper shows that the algebra generated by {C|  Aut(B_n)} is cyclic on H²(B_n), and any nonconstant function f  H²(B_n) is a cyclic vector of . In addition, the hypercyclic and cyclic composition operators will be discussed.     

Zeros in tables of characters for the groups Sn and An

In the representation theory of symmetric groups, for each partition of a natural number n, the partition h() of n is defined so as to obtain a certain set of zeros in the table of characters for S_n. Namely, h() is the greatest (under the lexicographic ordering ) partition among P(n) such that (g) 0. Here, is an irreducible character of S_n, indexed by a partition , and g is a conjugacy class of elements in S_n, indexed by a partition . We point out an extra set of zeros in the table that we are dealing with. For every non self-associated partition P(n), the partition f() of n is defined so that f() is greatest among the partitions of n which are opposite in sign to h() and are such that (g) 0 (Thm. 1). Also, for any self-associated partition of n > 1, we construct a partition () P(n) such that () is greatest among the partitions of n which are distinct from h() and are such that (g) 0 (Thm. 2).Supported by RFBR grant No. 04-01-00463 and by RFBR-BRFBR grant No. 04-01-81001.Translated from Algebra i Logika, Vol. 44, No. 1, pp. 24–43, January–February, 2005.     

Orthogonality of the characters of Sn

If the irreducible characters of the symmetric group are interpreted combinatorially using the Murnaghan-Nakayama formula, a sign reversing involution is given which proves the orthogonality formula of the first kind for these characters.     

Automorphisms of Chevalley groups of types Al, Dl, El over local rings without 1/2

In this paper, we prove that every automorphism of a Chevalley group of type B _l , l ≥ 2, over a commutative local ring with 1/2 is standard, i.e., it is a composition of ring, inner, and central automorphisms.     

Cut elimination for S4n and K4n with the central agent axiom

We analyze the multimodal logic S4 _n with the central agent axiom. We present a Hilbert-type calculus, then derive a Gentzen-type calculus with cut, and prove a cut-elimination theorem. The work shows that it is possible to construct a cut-free Gentzen-type calculus for this logic. Moreover, it also provides analogous results for the multimodal logic K4 _n with the central agent axiom.     

Non-neutrality of the Stiefel manifolds Vn,k II

The Stiefel manifolds V_2^m−1,k are shown to be non-neutral for m5, 2^m−1+2k=2ℓ<2^m−2.     

A combinatorial interpretation of the recurrence fn+1 = 6fn − fn−1

Bonin et al. (1993) recalled an open problem related to the recurrence relation verified by NSW numbers. The recurrence relation is the following: f_n+1 = 6f_n − f_n−1, with f₁ = 1 and f₂ = 7, and no combinatorial interpretation seems to be known. In this note, we define a regular language L whose number of words having length n is equal to f_n+1. Then, by using L we give a direct combinatorial proof of the recurrence.     

Reflection sieve obstructions in finite groups of type dn

In [4] we constructed certain homology representations of a finite group G of type A_n, B_n or C_n, and showed that these representations can be used to sift out the reflection compound characters of G. In the present note, we show that for a group G of type D_n, each reflection compound character π^(k), 2 k n − 2, determines a unique “obstruction” character θ^(k), which occurs with positive multiplicity in every homology representation containing π^(k).     

Invariants of the k-fold adjoint action of the Euclidean isometry group

A non-zero element of the Lie algebra \({\mathfrak{se}(3)}\) of the special Euclidean spatial isometry group SE(3) is known as a twist and the corresponding element of the projective Lie algebra is termed a screw. Either can be used to describe a one-degree-of-freedom joint between rigid components in a mechanical device or robot manipulator. This leads to a practical interest in multiple twists or screws, describing the overall instantaneous motion of such a device. In this paper, invariants of multiple twists under the action induced by the adjoint action of the group are determined. The ring of the polynomial invariants for the adjoint action of SE(3) acting on a single twist is well known to be finitely generated by the Klein and Killing forms, while a theorem of Panyushev (Publ. Res. Inst. Math. Sci. 4:1199–1257, 2007) gives finite generation for the real invariants of the induced action on two twists. However we are not aware of a corresponding theorem for k twists, where \({k\geq3}\). Following Study, Geometrie der Dynamen, (1903), we use the principle of transference to determine fundamental algebraic invariants and their syzygies. We prove that the ring of invariants for triple twists is rationally finitely generated by 13 of these invariants.     

Special uniform approximations of continuous vector-valued functions. Part II: special approximations in CX(T)CY(S)

In this paper, which is a continuation of Timofte (J. Approx. Theory 119 (2002) 291–299, we give special uniform approximations of functions from C_XY(T×S) and C_∞(T×S,XY) by elements of the tensor products C_X(T)C_Y(S), respectively C₀(T,X)C₀(S,Y), for topological spaces T,S and Γ-locally convex spaces X,Y (all four being Hausdorff).     

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.