论文标题
有限的一般线性组的传递性
Transitivity in finite general linear groups
论文作者
论文摘要
众所周知，置换组$ g $的瞬态子组的概念自然扩展到$ g $的子集。我们考虑一般线性组$ \ operatorName {gl}（n，q）$的子集，在类似国旗的结构上作用，这是$ \ mathbb {f} f} _q^n $的$ t $二维子空间的常见概括，$ t $ t $ t $ t $ t $ dimensional of $ t $ dimensional of $ t $ dimensional of $ t $ dimensional c。我们使用$ \ operatorname {gl}（gl}（n，q）$的$ \ operatorName {gl}（n，q）$的及传递子集的结构特征，并将其解释为$ \ propatatorname {gl}（n，q）$的共同类类协会方案中的设计类别。特别是，我们将perin定理概括为$ \ permatatorname {gl}（n，q）$在$ t $维二维子空间上进行传输的子组。 We survey transitive subgroups of $\operatorname{GL}(n,q)$, showing that there is no subgroup of $\operatorname{GL}(n,q)$ with $1<t<n$ acting transitively on $t$-dimensional subspaces unless it contains $\operatorname{SL}(n,q)$ or is one of two exceptional groups. On the other hand, for all fixed $t$, we show that there exist nontrivial subsets of $\operatorname{GL}(n,q)$ that are transitive on linearly independent $t$-tuples of $\mathbb{F}_q^n$, which also shows the existence of nontrivial subsets of $\operatorname{GL}(n,q)$ that are transitive on more general旗帜状结构。我们与正交多项式建立了连接，即al-salam-carlitz多项式，并通过rudvalis和shinoda概括了$ \ operatatorNamame {gl}（gl}（n，q）$的元素固定点的分布。我们的许多结果可以解释为对称组的相应结果的$ Q $ - 动物。
It is known that the notion of a transitive subgroup of a permutation group $G$ extends naturally to subsets of $G$. We consider subsets of the general linear group $\operatorname{GL}(n,q)$ acting transitively on flag-like structures, which are common generalisations of $t$-dimensional subspaces of $\mathbb{F}_q^n$ and bases of $t$-dimensional subspaces of $\mathbb{F}_q^n$. We give structural characterisations of transitive subsets of $\operatorname{GL}(n,q)$ using the character theory of $\operatorname{GL}(n,q)$ and interprete such subsets as designs in the conjugacy class association scheme of $\operatorname{GL}(n,q)$. In particular we generalise a theorem of Perin on subgroups of $\operatorname{GL}(n,q)$ acting transitively on $t$-dimensional subspaces. We survey transitive subgroups of $\operatorname{GL}(n,q)$, showing that there is no subgroup of $\operatorname{GL}(n,q)$ with $1<t<n$ acting transitively on $t$-dimensional subspaces unless it contains $\operatorname{SL}(n,q)$ or is one of two exceptional groups. On the other hand, for all fixed $t$, we show that there exist nontrivial subsets of $\operatorname{GL}(n,q)$ that are transitive on linearly independent $t$-tuples of $\mathbb{F}_q^n$, which also shows the existence of nontrivial subsets of $\operatorname{GL}(n,q)$ that are transitive on more general flag-like structures. We establish connections with orthogonal polynomials, namely the Al-Salam-Carlitz polynomials, and generalise a result by Rudvalis and Shinoda on the distribution of the number of fixed points of the elements in $\operatorname{GL}(n,q)$. Many of our results can be interpreted as $q$-analogs of corresponding results for the symmetric group.
