On the uniform spread of almost simple symplectic and orthogonal groups

On recognition of symplectic and orthogonal groups of small dimensions by spectrum

We refer to the set of the orders of elements of a finite group as its spectrum and say that finite groups are isospectral if their spectra coincide. In the paper we determine all finite groups isospectral to the simple groups $S_6(q)$, $O_7(q)$, and $O_8^+(q)$. In particular, we prove that with just four exceptions, every such a finite group is an extension of the initial simple group by a (possibly trivial) field automorphism.

Orthogonal and Symplectic Black Box Groups, Revisited

We propose a simple one sided Monte-Carlo algorithm to distinguish, to any given degree of certainty, between certain symplectic and orthogonal groups over fields of odd order. The algorithm does not use an order oracle and works in polynomial time.

A Probabilistic Approach to Conjugacy Classes in the Finite Symplectic and Orthogonal Groups

Markov chains are used to give a purely probabilistic way of understanding the conjugacy classes of the finite symplectic and orthogonal groups in odd characteristic. As a corollary of these methods one obtains a probabilistic proof of Steinberg's count of unipotent matrices and generalizations of formulas of Rudvalis and Shinoda.

Supercharacters of the Sylow p-subgroups of the finite symplectic and orthogonal groups

We define and study supercharacters of the classical finite unipotent groups of symplectic and orthogonal types (over any finite field of odd characteristic). We show how supercharacters for groups of those types can be obtained by restricting the supercharacter theory of the finite unitriangular group, and prove that supercharacters are orthogonal and provide a partition of the set of all irreducible characters. We also describe all irreducible characters of maximum degree in terms of the root system, and show how they can be obtained as constituents of particular supercharacters.

简单代数群在正交与辛 Grassmannians 上作用的通用稳定子

Generic stabilizers for simple algebraic groups acting on orthogonal and symplectic Grassmannians

本文研究简单代数群在自对偶不可约模及全奇异子空间簇上的忠实作用，证明了除有限情况外存在通用稳定子，并据此解决了经典群中具有稠密双陪集的极大连通子群对的确定问题。

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.

Noncommutative invariant theory of symplectic and orthogonal groups

We present a method for computing the Hilbert series of the algebra of invariants of the complex symplectic and orthogonal groups acting on graded noncommutative algebras with homogeneous components which are polynomial modules of the general linear group. We apply our method to compute the Hilbert series for different actions of the symplectic and orthogonal groups on the relatively free algebras of the varieties of associative algebras generated, respectively, by the Grassmann algebra and the algebra of $2\times 2$ upper triangular matrices. These two varieties are remarkable with the proper…

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…

A supercharacter theory for the Sylow p-subgroups of the finite symplectic and orthogonal groups

We define the superclasses for a classical finite unipotent group $U$ of type $B_{n}(q)$, $C_{n}(q)$, or $D_{n}(q)$, and show that, together with the supercharacters defined in a previous paper, they form a supercharacter theory. In particular, we prove that the supercharacters take a constant value on each superclass, and evaluate this value. As a consequence, we obtain a factorization of any superclass as a product of elementary superclasses. In addition, we also define the space of superclass functions, and prove that it is spanned by the supercharacters. As as consequence, we (re)obtain th…

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。

Powers in finite orthogonal and symplectic groups: A generating function approach

For an integer $M\geq 2$ and a finite group $G$, an element $α\in G$ is called an $M$-th power if it satisfies $A^M=α$ for some $A\in G$. In this article, we will deal with the case when $G$ is finite symplectic or orthogonal group over a field of order $q$. We introduce the notion of $M^*$-power SRIM polynomials. This, amalgamated with the concept of $M$-power polynomial, we provide the complete classification of the conjugacy classes of regular semisimple, semisimple, cyclic and regular elements in $G$, which are $M$-th powers, when $(M,q)=1$. The approach here is of generating functions, as…

Partial orthogonal spreads over $\mathbb{F}_2$ invariant under the symmetric and alternating groups

Let m be an integer greater than 2 and let V be a vector space of dimension 2^m over F_2. Let Q be a non-degenerate quadratic form of maximal Witt index defined on V. We show that the symmetric group S_{2m+1} acts on V as a group of isometries of Q and permutes the members of a partial orthogonal spread of size 2m+1. This implies that any group of even order 2m or odd order 2m+1 acts transitively and regularly on a partial orthogonal spread in V. We also show that the alternating group A_9 acts in a natural manner on a complete spread of size 9 defined on a vector space of dimension 8 over F_2…