Finite order elements in the integral symplectic group

整数辛群的公寓类

Apartment classes of integral symplectic groups

本文通过替代证明展示了辛群Sp_{2n}(Q)的Steinberg模块由整数公寓类生成。

Overgroups of elementary block-diagonal subgroups in the classical symplectic group over an arbitrary commutative ring

In this paper we prove a sandwich classification theorem for subgroups of the classical symplectic group over an arbitrary commutative ring $R$ that contain the elementary block-diagonal (or subsystem) subgroup $\operatorname{Ep}(ν, R)$ corresponding to a unitary equivalence realation $ν$ such that all self-conjugate equivalence classes of $ν$ are of size at least 4 and all not-self-conjugate classes of $ν$ are of size at least 5. Namely, given a subgroup $H$ of $\operatorname{Sp}(2n, R)$ such that $\operatorname{Ep}(ν, R) \le H$ we show that there exists a unique exact major form net of ideal…

Normality theorem for elementary symplectic group with respect to an alternating form

A.A. Suslin proved a normality theorem for an elementary linear group, which says that an elementary linear group of size bigger than or equal to 3 over a commutative ring with unity is normal in the general linear group of same size. Subsequently, V.I. Kopeiko extended this result of Suslin for a symplectic group defined with respect to the standard skew-symmetric matrix of even size. Here we generalise the result of Kopeiko for a symplectic group defined with respect to any invertible skew-symmetric matrix of even size of Pfaffian one.

Spectra of finite symplectic and orthogonal groups

The spectra of a finite group is the set of its element orders. We obtain an arithmetic description of finite symplectic and orthogonal groups. In particular, a description of spectra of all finite simple simplectic and orthogonal groups is obtained.

Symplectic structures on right-angled Artin groups: between the mapping class group and the symplectic group

We define a family of groups that include the mapping class group of a genus g surface with one boundary component and the integral symplectic group Sp(2g,Z). We then prove that these groups are finitely generated. These groups, which we call mapping class groups over graphs, are indexed over labeled simplicial graphs with 2g vertices. The mapping class group over the graph Gamma is defined to be a subgroup of the automorphism group of the right-angled Artin group A_Gamma of Gamma. We also prove that the kernel of the map Aut A_Gamma to Aut H_1(A_Gamma) is finitely generated, generalizing a th…

2n<250时辛群$Sp_{2n}(2)$的单奇异子群

Unisingular subgroups of symplectic group $Sp_{2n}(2)$ for $2n<250$

该研究围绕二元域上辛群的单奇异不可约子群存在性问题展开，解答了几乎所有2n<250次数下的对应问题，仅余7个n值待解，同时给出了单奇异不可约线性群结构的一般性结论。

Some remarks on the symplectic group $Sp(2g, \mathbb{Z})$

Let $G=\Sp(2g,\mathbb{Z})$ be the symplectic group over the integers. Given $m\in \mathbb{N}$, it is natural to ask if there exists a non-trivial matrix $A\in G$ such that $A^{m}=I$, where $I$ is the identity matrix in $G$. In this paper, we determine the possible values of $m\in \mathbb{N}$ for which the above problem has a solution. We also show that there is an upper bound on the maximal order of an element in $G$. As an illustration, we apply our results to the group $\Sp(4,\mathbb{Z})$ and determine the possible orders of elements in it. Finally, we use a presentation of $\Sp(4,\mathbb{Z}…

Products of two involutions in orthogonal and symplectic groups

An element of a group is called bireflectional when it is the product of two involutions of the group (i.e. elements of order 1 or 2). If an element is bireflectional then it is conjugated to its inverse. It is known that all elements of orthogonal groups of quadratic forms are bireflectional. F. Bünger has characterized the elements of unitary groups (over fields of characteristic not $2$) that are bireflectional. Yet in symplectic groups over fields with characteristic different from 2, in general there are elements that are conjugated to their inverse but are not bireflectional (however, ov…

特征为2的有限域上辛李代数中的交换对

Commuting pairs in symplectic Lie algebras over finite fields of characteristic two

参照奇特征下的相关研究，给出特征为2的有限域上辛李代数中有序交换对个数的生成函数。

正交群与辛群中指数为$2$的幂幺元的乘积

Products of unipotent elements of index $2$ in orthogonal and symplectic groups

该研究刻画了正交群与辛群中可表为两个指数2的幂幺等距元乘积的元素，给出辛群情形的充要条件，证明辛群元素均可表为三个指数2幂幺元的乘积，并指出正交群相关分类与平方零斜自伴算子和的分类高度吻合。

On the Conjugacy Classes in the orthogonal and symplectic groups over algebraically closed fields

Let $\F$ be an algebraically closed field. Let $\V$ be a vector space equipped with a non-degenerate symmetric or symplectic bilinear form $B$ over $\F$. Suppose the characteristic of $\F$ is \emph{large}, i.e. either zero or greater than the dimension of $\V$. Let $I(\V, B)$ denote the group of isometries. Using the Jacobson-Morozov lemma we give a new and simple proof of the fact that two elements in $I(\V,B)$ are conjugate if and only if they have the same elementary divisors.

Canonical Symplectic Representations for Prime Order Conjugacy Classes of the Mapping-class Group

In this paper we find a unique normal form for the symplectic matrix representation of the conjugacy class of a prime order element of the mapping-class group. We find a set of generators for the fundamental group of a surface with a conformal automorphism of prime order which reflects the action the automorphism in an optimal way. This is called an {\sl adapted} homotopy basis and there is a corresponding {\sl adapted presentation}. We also give a necessary and sufficient condition for a prime order symplectic matrix to be the image of a prime order element in the mapping-class group.