Linear spaces with significant characteristic prime

正则图的分类：具有欧拉特征数$-p^4$的正则图分类

A classification of regular maps with Euler characteristic $-p^4$ for a prime $p\geq 5$

本文研究了欧拉特征数为$-p^4$的正则图，证明了其Sylow $p$-子群的阶数限制，并分类了所有此类正则图，表明当$p$不在{2,3,5,7,13}时，相关曲面不支持正则图。

素特征域上典型单李代数的循环群分次

Gradings by cyclic groups on classical simple Lie algebras in prime characteristics

该研究分类了素特征域上典型单李代数的有限循环群分次，将其转化为Weyl型群的轨道问题，推广了Kac坐标分类并明确了特征p下的限制条件。

Lower bounds on the maximum dimension of a simple module in characteristic p

We obtain lower bounds for the maximum dimension of a simple FG-module, where G is a finite group and F is an algebraically closed field of characteristic p. The bounds are described in terms of properties of p-subgroups of G. When p is 2 or p is a Mersenne prime, the bounds take a different form, due to exceptions which arise for such primes.

On embeddings of certain spherical homogeneous spaces in prime characteristic

Let $\mc G$ be a reductive group over an algebraically closed field of characteristic $p>0$. We study homogeneous $\mc G$-spaces that are induced from the $G\times G$-space $G$, $G$ a suitable reductive group, along a parabolic subgroup of $\mc G$. We show that, under certain mild assumptions, any (normal) equivariant embedding of such a homogeneous space is canonically Frobenius split compatible with certain subvarieties and has an equivariant rational resolution by a toroidal embedding. In particular, all these embeddings are Cohen-Macaulay. Examples are the $G\times G$-orbits in normal redu…

奇正特征下的不可分裂的锐2传递群

Non-split sharply 2-transitive groups of odd positive characteristic

本文构造了奇正特征下首个不可分裂的锐2传递群，并证明无2阶挠元的群可嵌入此类群，从而在足够大的特征下得到大量不同构的例子。

The number of composition factors of order $p$ in completely reducible groups of characteristic $p$

Let $q$ be a power of a prime $p$ and let $G$ be a completely reducible subgroup of $\mathrm{GL}(d,q)$. We prove that the number of composition factors of $G$ that have prime order $p$ is at most $(\varepsilon_q d-1)/(p-1)$, where $\varepsilon_q$ is a function of $q$ satisfying $1\leqslant\varepsilon_q\leqslant 3/2$. For every $q$, we give examples showing this bound is sharp infinitely often.

On the Decomposition Numbers of the Ree Groups 2F4(q^2) in Non-Defining Characteristic

We compute the l-modular decomposition matrices of the simple Ree groups 2F4(q^2), where q^2=2^{2n+1} and n is a positive integer, for all primes l > 3 up to some entries in the unipotent characters. Using these matrices we determine the smallest degree of a non-trivial irreducible l-modular representation of 2F4(q^2) for all primes l > 3. We also obtain results on the 3-modular decomposition matrices of 2F4(q^2).

Conjugacy classes in Sylow p-subgroups of finite Chevalley groups in bad characteristic

Let $U = \mathbf U(q)$ be a Sylow $p$-subgroup of a finite Chevalley group $G = \mathbf G(q)$. In [GR}] Röhrle and the second author determined a parameterization of the conjugacy classes of $U$, for $\mathbf G$ of small rank when $q$ is a power of a good prime for $\mathbf G$. As a consequence they verified that the number $k(U)$ of conjugacy classes of $U$ is given by a polynomial in $q$ with integer coefficients. In the present paper, we consider the case when $p$ is a bad prime for $\mathbf G$. We obtain a parameterization of the conjugacy classes of $U$, when $\mathbf G$ has rank less tha…

A characteristic subgroup for fusion systems

As a counterpart for the prime 2 to Glauberman's $ZJ$-theorem, Stellmacher proves that any nontrivial 2-group $S$ has a nontrivial characteristic subgroup $W(S)$ with the following property. For any finite $Σ_4$-free group $G$, with $S$ a Sylow 2-subgroup of $G$ and with $O_2(G)$ self-centralizing, the subgroup $W(S)$ is normal in $G$. We generalize Stellmacher's result to fusion systems. A similar construction of $W(S)$ can be done for odd primes and gives rise to a Glauberman functor.