Finite symplectic matrix groups

Finite order elements in the integral symplectic group

For $g\in \mathbb{N}$, let $G=\Sp(2g,\mathbb{Z})$ be the integral symplectic group and $S(g)$ be the set of all positive integers which can occur as the order of an element in $G$. In this paper, we show that $S(g)$ is a bounded subset of $\mathbb{R}$ for all positive integers $g$. We also study the growth of the functions $f(g)=|S(g)|$, and $h(g)=max\{m\in \mathbb{N}\mid m\in S(g)\}$ and show that they have at least exponential growth.

Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type II. Unipotent classes in symplectic groups

We show that Nichols algebras of most simple Yetter-Drinfeld modules over the projective symplectic linear group over a finite field, corresponding to unipotent orbits, have infinite dimension. We give a criterium to deal with unipotent classes of general finite simple groups of Lie type and apply it to regular classes

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.

Classification of subgroups of symplectic groups over finite fields containing a transvection

In this note we give a self-contained proof of the following classification (up to conjugation) of subgroups of the general symplectic group of dimension n over a finite field of characteristic l, for l at least 5, which can be derived from work of Kantor: G is either reducible, symplectically imprimitive or it contains Sp(n, l). This result is for instance useful for proving "big image" results for symplectic Galois representations.

On the Weil representation of infinite-dimensional symplectic group over a finite field

We extend the Weil representation of infinite-dimensional symplectic group to a representation a certain category of linear relations.

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.

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.

Finite groups of symplectic automorphisms of K3 surfaces in positive characteristic

We show that Mukai's classification of finite groups which may act symplectically on a complex K3 surface extends to positive characteristic $p$ under the assumptions that (i) the order of the group is coprime to $p$ and (ii) either the surface or its quotient is not birationally isomorphic to a supersingular K3 surface with Artin invariant 1. In the case without the assumption (ii) we classify all possible new groups which may appear. We prove that the assumption on the order of the group is always satisfied if $p > 11$ and if $p=2,3,5,11$ we give examples of K3 surfaces with finite symplecti…

奇特征下超奇异 K3 曲面的辛自同构有限群

Finite Groups of Symplectic Automorphisms of Supersingular K3 surfaces in Odd Characteristics

本文利用格理论方法研究奇特征下有限群在 K3 曲面上的辛作用，提出 p-根对机制约束超奇异 K3 曲面的非温顺辛作用，并给出群阶中 p 的指数上界。

A family of non-FSZ finite symplectic groups

Let $p$ be an odd prime with $p\equiv1\bmod 4$. Then for any odd power $q$ of $p$ and a positive integer $j$ we show that the groups $\text{Sp}_{p^j+1}(q),\text{PSp}_{p^j+1}(q)$, and their Sylow $p$-subgroups are non-$FSZ_{p^j}$.

The Schur multiplier of finite symplectic groups

We show that the Schur multiplier of $Sp(2g,\mathbb Z/D\mathbb Z)$ is $\mathbb Z/2\mathbb Z$, when $D$ is divisible by 4.

Real representations of finite symplectic groups over fields of characteristic two

We prove that when $q$ is a power of $2$, every complex irreducible representation of $\mathrm{Sp}(2n, \mathbb{F}_q)$ may be defined over the real numbers, that is, all Frobenius-Schur indicators are 1. We also obtain a generating function for the sum of the degrees of the unipotent characters of $\mathrm{Sp}(2n, \mathbb{F}_q)$, or of $\mathrm{SO}(2n+1, \mathbb{F}_q)$, for any prime power $q$.