Linear Operators Preserving Symplectic Group over a Field Consisting of at Least Four Elements

Suppose F is a field consisting of at least four elements. Let Mn（F） and SP2n（F） be the linear space of all n × n matrices and the group of all 2n × 2n symplectic matrices over F, respectively. A linear operator L ： M2n（F） → M2n（F） is said to preserve the symplectic group if L（SP2n（F）） = SP2n（F）. It is shown that L is an invertible preserver of the symplectic group if and only if L takes the form （i） L（X） = QPXP^-1 for any X ∈ M2n（F） or （ii） L（X） = QPX^TP^-1 for any X ∈M2n（F）, where Q ∈ SP2n（F） and P is a generalized symplectic matrix. This generalizes the result derived by Pierce in Canad J. Math., 3（1975）, 715-724.

The Essential 2-rank for Classical Groups

Let G be a symplectic or orthogonal group defined over a finite field with odd characteristic and let D≤G be a Sylow 2-subgroup.In this paper,we classify the essential 2-subgroups and determine the essential 2-rank of the Frobenius category FD(G).Together with the results of An–Dietrich and Cao–An–Zeng,this completes the work of essential subgroups and essential ranks of classical groups.

Lattices Generated by Joins of the Flats in Orbits under Finite Affine-singular Symplectic Group and its Characteristic Polynomials

Let ASG（2v ＋ l, v; Fq） be the （2v ＋l）-dimensional affine-singular symplectic space over the finite field Fq and ASp2v＋l,v（Fq） be the affine-singular symplectic group of degree 2v ＋ l over Fq. Let O be any orbit of flats under ASp2v＋l,v（Fq）. Denote by J the set of all flats which are joins of flats in O such that O LJ and assume the join of the empty set of flats in ASG（2v ＋ l, v;Fq） is φ. Ordering LJ by ordinary or reverse inclusion, then two lattices axe obtained. This paper firstly studies the inclusion relations between different lattices, then determines a characterization of flats contained in a given lattice LJ, when the lattices form geometric lattice, lastly gives the characteristic polynomial of LJ.

ON HARMONIC MAPS INTO SYMPLECTIC GROUPS Sp(N)

By means of the theory of harmonic maps into the unitary group U(N), the authors study harmonic maps into the symplectic group Sp(N). The symplectic uniton and symplectic ex--tended uniton are introduced. The method of the symplectic Backlund transformation and the Darboux transformation is used to construct new symplectic unitons from a known one.

Topological Structures of ω-subsets in Symplectic Groups

All the symplectic matrices possessing a fixed eigenvalue ω on the unit circle form a hypersurface in the real symplectic group Sp(2n). This paper is devoted to the study of the topological structures of this hypersurface and its complement in Sp(2n).

A Recursive Basis for Primitive Forms in Symplectic Spaces and Applications to Heisenberg Groups

This paper is divided in two parts： in Section 2, we define recursively a privileged basis of the primitive forms in a symplectic space（V^（2n）, ω）. Successively, in Section 3, we apply our construction in the setting of Heisenberg groups H^n, n ≥ 1, to write in coordinates the exterior differential of the so-called Rumin＇s complex of differential forms in H^n.

SO^(*)(2n)的极小和限制极小幂零轨道的维数

在交换对合下,讨论了so^(*)(2n)的余伴随轨道和限制性极小幂零轨道,并得到了其对应的轨道维数.此外,在最高根意义下,建立了基本余伴随轨道维数与极小幂零轨道维数之间的关系.

Full automorphism group of the generalized symplectic graph

Let Fq be a finite field of odd characteristic, m, v the integers with 1 ≤ m ≤ v and K a 2v × 2v nonsingular alternate matrix over Fq. In this paper, the generalized symplectic graph GSp2v（q, m） relative to K over Fq is introduced. It is the graph with m-dimensional totally isotropic subspaces of the 2v-dimensional symplectic space Fq（2v） as its vertices and two vertices P and Q are adjacent if and only if the rank of PKQw is 1 and the dimension of P ∩ Q is m - 1. It is proved that the full automorphism group of the graph GSp2v（q, m） is the projective semilinear symplectic group P∑p（2v, q）.

On symplectic automorphisms of elliptic surfaces acting on CH_(0)

Let S be a complex smooth projective surface of Kodaira dimension one. We show that the group Auts(S) of symplectic automorphisms acts trivially on the Albanese kernel CH_(0)(S)albof the 0-th Chow group CH_(0)(S), unless possibly if the geometric genus and the irregularity satisfy pg(S) = q(S) ∈ {1, 2}. In the exceptional cases, the image of the homomorphism Auts(S) → Aut(CH_(0)(S)alb) has the order at most 3. Our arguments actually take care of the group Autf(S) of fibration-preserving automorphisms of elliptic surfaces f : S → B. We prove that if σ ∈ Autf(S) induces the trivial action on Hi,0(S) for i > 0, then it induces the trivial action on CH_(0)(S)alb. As a by-product we obtain that if S is an elliptic K3 surface, then Autf(S)∩Auts(S)acts trivially on CH_(0)(S)alb.

代数几何和表示论的一些新进展

双有理几何在双有理等价意义下对射影代数簇及其推广射影偶进行分类,是代数几何中的一个主要研究方向.代数几何与表示论具有不可分割的紧密联系,辛奇点就是同时涉及到这两个领域的一个重要研究课题.本文在对双有理几何中的基本概念进行简短的介绍后,探讨了典范体积的分布及应用,该不变量在射影偶的有界性问题上发挥关键作用.介绍了辛奇点的形变及量子化,及其在表示论中的应用.

Conjugacy Classes of Torsion in 4×4 Integral Symplectic Group

A complete list of representatives of conjugacy classes of torsion in 4×4 integral symplectic group is given in this paper. There are 55 distinct such classes and each torsion element has order of 2, 3, 4, 5, 6, 8, 10 and 12.

The diagram category of framed tangles and invariants of quantized symplectic group

In this paper,we present a categorical version of the first and second fundamental theorems of the invariant theory for the quantized symplectic groups.Our methods depend on the theory of braided strict monoidal categories which are pivotal,more explicitly,the diagram category of framed tangles.

Finite Group Action and Gromov-Witten Invariants in Symplectic Four-Manifolds

Let a finite group act semi-freely on a closed symplectic four-manifold with a 2-dimensional fixed point set. Then we show that the relative Gromov-Witten invariants are the same as the invariants on the quotient set-up with respect to the fixed point set.

欧氏环上辛群在线性群中的扩群

论文定出了欧氏环上辛群在线性群中的全部扩群 ,得到如下结果 :设R是欧氏环 ,N =Sp(2m ,R) ,G=GL(2m ,R) ,N≤X≤G .则存在R的一个理想S ,使X NS=g∈SL(2m ,R)fS(g)∈Sp(2m ,R/S) ,其中fS:SL(m ,R) →SL(m ,R/S)

基于辛空间的具有仲裁的认证码的构造

具有仲裁的认证码既要防止敌手的欺骗，又要防止收方和发方的互相欺骗．该文给出一种由辛几何构造具有仲裁的认证码的方法，并计算了有关参数。

多体动力学的几何积分方法研究进展

动力系统的几何积分研究是近20年来工程计算领域非常活跃的方向.多体动力学方程(微分方程,微分代数方程)是一类典型的动力系统,将其从Lagrange体系向Hamilton系统过渡,目的在于从欧氏几何过渡到辛几何形态,将对偶变量引入到力学研究中,然后利用辛几何的数学框架对多体系统动力学方程进行数值计算,可以预知多体动力学系统的一些定性信息,并在数值离散时能保持这些定性性质特征,尤其在表示关键的物理意义时需要强调保持这些几何性质.简要介绍多体系统(无约束多刚体系统、完整约束多刚体系统和柔性多体系统)的Hamilton正则方程的建立和几何积分方法的构造,着重介绍了在多体动力学计算中非常有应用前景的高阶辛算法(合成辛算法、分裂合成辛算法和辛精细积分法)、多辛算法,以及广义Hamilton系统与Lie群积分方法等计算几何力学方法,并对Lie群积分的投影方法、流形局部坐标法等方法进行了阐述.

奇异典型群作用下子空间轨道的长度

设q是q个元素的有限域,qn+l是Fq上n+l维行向量空间,Gn+l,n是n+l级奇异典型群之一.qn+l在Gn+l,n上的作用下导出了它在qn+l的子空间集合上的作用,因而qn+l在Gn+l,n作用下划分成一些轨道Mn+l,n.采用矩阵初等行变换的方法,分别给出奇异辛群,奇异酉群作用下子空间轨道Mn+l,n的长度.

有限奇异辛群作用下轨道中子空间的和生成的格(Ⅰ)(英文)

设Fq(2ν+l)是有限域Fq上的(2ν+l)-维向量空间,Sp2ν+l,ν(Fq)是Fq上2ν+l级奇异辛群,M为Sp2ν+l,ν(Fq)作用下的任一子空间轨道.LJ表示M中子空间的和的集合,并假定Fq(2ν+l)的0个子空间的和是{0}子空间,按包含或反包含关系来定义LJ的偏序,可得两个格.研究了不同格之间的包含关系,含于一个给定的格LJ中的子空间的特征以及格LJ的特征多项式.

局部环上辛群在线性群中的扩群

本文完全定出了局部环上辛群在一般线性群中的扩群.

面向油气勘探开发提升地震偏移及属性刻划水平

针对复杂地质条件油气勘探开发的需求 ,仅就运用地震资料提取地质属性及构建油储模型所可能面临的难点 ,阐明必需提升地震波传播及成像研究水平的基础理论命题和数值计算的关键环节 .文中着重于用形象的数值图像结果 ,说明开展此类研究的必要性 .对我国而言 ,前新生代海相碳酸盐岩勘探和今后相当时期内对国民经济仍起支撑作用的东部老油田外围及深部油气的勘探开发 ,这两类研究对象尽管其地质演化年代及过程各异 ,但对地球物理偏移成像技术进步及油储建模而言 ,对提升探测的理论所提出的要求 ,都属于近代的难题 .