Let $s$ be an $n$-dimensional symplectic form over a field $\mathbb{F}$ of characteristic other than $2$, with $n>2$. In a previous article, we have proved that if $\mathbb{F}$ is infinite …
Let $s$ be an $n$-dimensional symplectic form over a field $\mathbb{F}$ of characteristic other than $2$, with $n>2$. In a previous article, we have proved that if $\mathbb{F}$ is infinite then every element of the symplectic group $\mathrm{Sp}(s)$ is the product of four involutions if $n$ is a multiple of $4$ and of five involutions otherwise. Here, we adapt this result to all finite fields with characteristic not $2$, with the sole exception of the very special situation where $n=4$ and $|\mathbb{F}|=3$, a special case which we study extensively.
Let $s$ be an $n$-dimensional symplectic form over an arbitrary field with characteristic not $2$, with $n>2$. The simplicity of the group $\mathrm{Sp}(s)/\{\pm \mathrm{id}\}$ and the existence of a non-trivial …
Let $s$ be an $n$-dimensional symplectic form over an arbitrary field with characteristic not $2$, with $n>2$. The simplicity of the group $\mathrm{Sp}(s)/\{\pm \mathrm{id}\}$ and the existence of a non-trivial involution in $\mathrm{Sp}(s)$ yield that every element of $\mathrm{Sp}(s)$ is a product of involutions. Extending and improving recent results of Awa, de La Cruz, Ellers and Villa with the help of a completely new method, we prove that if the underlying field is infinite, every element of $\mathrm{Sp}(s)$ is the product of four involutions if $n$ is a multiple of $4$, and of five involutions otherwise. The first part of this result is shown to be optimal for all multiples of $4$ and all fields, and is shown to fail for the fields with three elements and for $n=4$. Whether the second part of the result is optimal remains an open question. Finite fields will be tackled in a subsequent article.
Let In be the n×n identity matrix and J=0In−In0. A matrix A is called symplectic if ATJA=J. A symplectic matrix A is a commutator of symplectic involutions if A=XYX−1Y−1, where …
Let In be the n×n identity matrix and J=0In−In0. A matrix A is called symplectic if ATJA=J. A symplectic matrix A is a commutator of symplectic involutions if A=XYX−1Y−1, where X and Y are symplectic matrices satisfying X2=Y2=I. In this article, we give necessary and sufficient condition for a symplectic matrix over the complex number field to be expressed as a product of two commutators of symplectic involutions.
在斜地 K 上认为稳定的 Steinberg 组圣是(K) 。元素 x 被称为复杂物如果 x~2 = 1。在这篇论文，复杂物被允许是身份。Theauthors 证明直到变化形式的 GL_n (K) 的元素 A 能作为 BC 被代表，在 B 更低的地方三角形并且 C 同时是上面的三角形。而且， B 和 C 能被选择以便在 B 的主要对角线的元素是β _ …
在斜地 K 上认为稳定的 Steinberg 组圣是(K) 。元素 x 被称为复杂物如果 x~2 = 1。在这篇论文，复杂物被允许是身份。Theauthors 证明直到变化形式的 GL_n (K) 的元素 A 能作为 BC 被代表，在 B 更低的地方三角形并且 C 同时是上面的三角形。而且， B 和 C 能被选择以便在 B 的主要对角线的元素是β _ 1，β _ 2，吗？ ? ?，β _ n，并且照顾γ _ 1，γ _ 2，？？，γ _ n c_n c_n ∈[K~* ， K~*] 并且Π _(j=1 )~ n β _ j γ _ j = detA。在圣(K) 的每元素δ是 10 复杂物的一个产品，这也被证明。
For a nontrivial additive character A and a multiplicative character X of the finite field with q elements, the 'Gauss' sums ~A(trg) over g ESp(2n, q) and ~x(detg)A(trg) over g …
For a nontrivial additive character A and a multiplicative character X of the finite field with q elements, the 'Gauss' sums ~A(trg) over g ESp(2n, q) and ~x(detg)A(trg) over g C GSp(2n, q) are considered. We show that it can be expressed as a polynomial in q with coefficients involving powers of Kloosterman sums for the first one and as that with coefficients involving sums of twisted powers of Kloosterman sums for the second one. As a result, we can determine certain 'generalized Kloosterman sums over nonsingular matrices' and 'generalized Kloosterman sums over nonsingular alternating matrices', which were previously determined by J. H. Hodges only in the case that one of the two arguments is zero.
Let $R$ be a commutative ring, and let $l\ge 2$; for $l=2$ it is assumed additionally that $R$ has no residue fields of two elements. The subgroups of the general …
Let $R$ be a commutative ring, and let $l\ge 2$; for $l=2$ it is assumed additionally that $R$ has no residue fields of two elements. The subgroups of the general linear group $\operatorname {GL}(n,R)$ that contain the elementary symplectic group $\operatorname {Ep}(2l,R)$ are described. In the case where $R=K$ is a field, similar results were obtained earlier by Dye, King, and Shang Zhi Li.
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 …
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. B\"unger 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, over fields of characteristic 2, every element of a symplectic group is bireflectional). In this article, we characterize the bireflectional elements of symplectic groups in terms of Wall invariants, over fields of characteristic not 2: the result is cited without proof in B\"unger's PhD thesis, and attributed to Klaus Nielsen. We also take advantage of our approach to give a simplified proof of Wonenburger's corresponding result for orthogonal groups and general linear groups.
This paper is concerned with two mani problems: (a) the determination of the conjugacy classes in the finite-dimensional unitary, symplectic and orthogonal groups over division rings or fields ; (b) …
This paper is concerned with two mani problems: (a) the determination of the conjugacy classes in the finite-dimensional unitary, symplectic and orthogonal groups over division rings or fields ; (b) the determination of the equivalence classes of non-degenerate sesquilinear forms on finite-dimensional vector spaces .