Generation of classical groups over finite fields

Let U_n(V) and Sp_n(V) denote the unitary group and the symplectic group of the n dimensional vector space V over a finite field of characteristic not 2, respectively. Assume that the hyperbolic rank of U_n(V) is at least one. Then U_n(V) is generated by 4 elements and Sp_n(V) by 3 elements. Further, U_2m+1(V) is generated by 3 elements and Sp_4m(V) by 2 elements.     

Automorphisms of the symplectic group over a local ring

Let R denote a commutative local ring with maximal ideal m and residue field K = R/m. Let V be a symplectic space over R. In this paper we determine the group automorphisms of the symplectic group Sp_n(V) when n 6, the characteristic of k is not 2, and k is not the finite field of three elements.     

On the trivectors of a 6-dimensional symplectic vector space

Let V be a 6-dimensional vector space over a field F, let f be a nondegenerate alternating bilinear form on V and let Sp(V,f)≅Sp₆(F) denote the symplectic group associated with (V,f). The group GL(V) has a natural action on the third exterior power ?³V of V and this action defines five families of nonzero trivectors of V (four of whose are orbits for any choice of F). In this paper, we divide three of these five families into orbits for the action of Sp(V,f)⊆GL(V) on ?³V.     

Generators of Spn(V) over a quasi semilocal semihereditary ring

This paper is devoted to determine the minimal length of expressions of an isometry in a symplectic group Sp_n(V) by a product of transvections under the assumption that V is an n-ary nonsingular alternating space over a quasi semilocal semihereditary ring with 2 as a unit.     

A 14-dimensional Module for the Symplectic Group: Orbits on Vectors

Let 𝔽 be a field, V a 6-dimensional 𝔽-vector space and f a nondegenerate alternating bilinear form on V. We consider a 14-dimensional module for the symplectic group Sp(V, f) ? Sp(6, 𝔽) associated with (V, f), and classify the orbits on vectors. For characteristic distinct from 2, this module is irreducible and isomorphic to the Weyl module of Sp(V, f) for the fundamental weight λ₃. If the characteristic is 2, then the module is reducible as it contains an 8-dimensional submodule isomorphic to the spin module of Sp(V, f).     

First Fundamental Theorem for Covariants of Classical Groups

Let U(G) be a maximal unipotent subgroup of one of the classical groups G=GL(V), O(V), Sp(V). Let W be a direct sum of copies of V and its dual V*. For the natural action U(G) : W, we describe a minimal system of homogeneous generators for the algebra of U(G)-invariant regular functions on W. For G=O(V), Sp(V), this result is connected with a construction for the irreducible representations of G due to H. Weyl.     

Poisson deformations of symplectic quotient singularities

We establish a connection between smooth symplectic resolutions and symplectic deformations of a (possibly singular) affine Poisson variety.In particular, let V be a finite-dimensional complex symplectic vector space and G⊂Sp(V) a finite subgroup. Our main result says that the so-called Calogero-Moser deformation of the orbifold V/G is, in an appropriate sense, a versal Poisson deformation. That enables us to determine the algebra structure on the cohomology of any smooth symplectic resolution X?V/G (multiplicative McKay correspondence). We prove further that if is an irreducible Weyl group and , then no smooth symplectic resolution of V/G exists unless G is of types .     

On the Buchstaber Subring in MS p

A formula is given to calculate the last n number of symplectic characteristic classes of the tensor product of the vector Spin(3)- and Sp(n)-bundles through its first 2n number of characteristic classes and through characteristic classes of Sp(n)-bundle. An application of this formula is given in symplectic cobordisms and in rings of symplectic cobordisms of generalized quaternion groups.     

On the some normal subgroups of the extended modular group

In this paper, we give the group structures and the signatures of some normal subgroups of the extended modular group Π containing the principal congruence subgroup Γ(12).     

On the pronormality of subgroups of odd index in finite simple symplectic groups

A subgroup H of a group G is pronormal if the subgroups H and H ^g are conjugate in 〈H,H ^g 〉 for every g ∈ G. It was conjectured in [1] that a subgroup of a finite simple group having odd index is always pronormal. Recently the authors [2] verified this conjecture for all finite simple groups other than PSL _n (q), PSU _n (q), E ₆(q), ² E ₆(q), where in all cases q is odd and n is not a power of 2, and P Sp_2n (q), where q ≡ ±3 (mod 8). However in [3] the authors proved that when q ≡ ±3 (mod 8) and n ≡ 0 (mod 3), the simple symplectic group P Sp²ⁿ (q) has a nonpronormal subgroup of odd index, thereby refuted the conjecture on pronormality of subgroups of odd index in finite simple groups.The natural extension of this conjecture is the problem of classifying finite nonabelian simple groups in which every subgroup of odd index is pronormal. In this paper we continue to study this problem for the simple symplectic groups P Sp_2n (q) with q ≡ ±3 (mod 8) (if the last condition is not satisfied, then subgroups of odd index are pronormal). We prove that whenever n is not of the form 2 ^m or 2 ^m (2^2k +1), this group has a nonpronormal subgroup of odd index. If n = 2 ^m , then we show that all subgroups of P Sp_2n (q) of odd index are pronormal. The question of pronormality of subgroups of odd index in P Sp_2n (q) is still open when n = 2 ^m (2^2k + 1) and q ≡ ±3 (mod 8).     

Harmonic morphisms from the classical non-compact semisimple Lie groups

We construct the first known complex-valued harmonic morphisms from the non-compact Lie groups SLn(R), SU^∗(2n) and Sp(n,R) equipped with their standard Riemannian metrics. We then introduce the notion of a bi-eigenfamily and employ this to construct the first known solutions on the non-compact Riemannian SO^∗(2n), SO(p,q), SU(p,q) and Sp(p,q). Applying a duality principle we then show how to manufacture the first known complex-valued harmonic morphisms from the compact Lie groups SO(n), SU(n) and Sp(n) equipped with semi-Riemannian metrics.     

Maslov Index and a Central Extension of the Symplectic Group

In this paper we show that over any field K of characteristic different from 2, the Maslov index gives rise to a 2-cocycle on the stable symplectic group with values in the Witt group. We also show that this cocycle admits a natural reduction to I ²(K) and that the induced natural homomorphism from K ₂ Sp(K)I ²(K) is indeed the homomorphism given by the symplectic symbol {x, y} mapping to the Pfister form 1, -x 1, –y.     

On Odd Symplectic Schur Functions

Proctor defined combinatorially a family of Laurent Polynomials, called odd symplectic Schur functions, indexed by pairs (λ, c), where λ is partition andcis a column length of λ. A conjecture of Proctor (Invent. Math.92,1988, 307–332) includes the statement that the odd symplectic Schur functions are actually characters ofSp(2n + 1, C). The purpose of the present note is to prove this.     

On the semiproportional character conjecture in groups Sp<Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

Previously, the author made the following conjecture: if a finite group has two semiproportional irreducible characters φ and ψ, then φ(1) = ψ(1). In the present paper, a new confirmation of the conjecture is obtained. Namely, the conjecture is verified for symplectic groups Sp₄(q) and PSp₄(q).     

GW invariants relative to normal crossing divisors

In this paper we introduce a notion of symplectic normal crossing divisor V and define the GW invariant of a symplectic manifold X relative to such a divisor. Our definition includes normal crossing divisors from algebraic geometry. The invariants we define in this paper are key ingredients in symplectic sum type formulas for GW invariants, and extend those defined in our previous joint work with T.H. Parker [16], which covered the case V   was smooth. The main step is the construction of a compact moduli space of relatively stable maps into the pair (X,V)$(X,V)$ in the case V is a symplectic normal crossing divisor in X.     

Subgroups of GL 1(R) for Local Rings R

Abstract

Let R be a local ring, with maximal ideal m , and residue class division ring R/ m ?=?D. Denote by R*?=?G L ₁(R), the group of units of R. Here we investigate some algebraic structure of subnormal and maximal subgroups of R*. For instance, when D is of finite dimension over its center, it is shown that finitely generated subnormal subgroups of R* are central. It is also proved that maximal subgroups of R* are not finitely generated. Furthermore, assume that P is a nonabelian maximal subgroup of R* such that P contains a noncentral soluble normal subgroup of finite index, it is shown that D is a crossed product division algebra.     

The Classification of Symplectic Structures of Convex Type

In this paper we give a classification of a certain class of semisimple symplectic structures, more precisely all symplectic structures for which a symplectic module (V,) is of convex type. This classification then leads to a classification of Lie algebras with invariant cones and at most one dimensional center.