Transitivity in finite general linear groups

IF 1 3区 数学 Q1 MATHEMATICS Mathematische Zeitschrift Pub Date : 2024-05-30 DOI:10.1007/s00209-024-03511-x
Alena Ernst, Kai-Uwe Schmidt
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Abstract

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 \({{\,\textrm{GL}\,}}(n,q)\) 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 \({{\,\textrm{GL}\,}}(n,q)\) using the character theory of \({{\,\textrm{GL}\,}}(n,q)\) and interpret such subsets as designs in the conjugacy class association scheme of \({{\,\textrm{GL}\,}}(n,q)\). In particular we generalise a theorem of Perin on subgroups of \({{\,\textrm{GL}\,}}(n,q)\) acting transitively on t-dimensional subspaces. We survey transitive subgroups of \({{\,\textrm{GL}\,}}(n,q)\), showing that there is no subgroup of \({{\,\textrm{GL}\,}}(n,q)\) with \(1<t<n\) acting transitively on t-dimensional subspaces unless it contains \({{\,\textrm{SL}\,}}(n,q)\) or is one of two exceptional groups. On the other hand, for all fixed t, we show that there exist nontrivial subsets of \({{\,\textrm{GL}\,}}(n,q)\) that are transitive on linearly independent t-tuples of \(\mathbb {F}_q^n\), which also shows the existence of nontrivial subsets of \({{\,\textrm{GL}\,}}(n,q)\) 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 \({{\,\textrm{GL}\,}}(n,q)\). Many of our results can be interpreted as q-analogs of corresponding results for the symmetric group.

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有限一般线性群中的传递性
我们考虑一般线性群 ({{\textrm{GL}\,}}(n,q)\)的子集在旗状结构上的传递作用,这些结构是 \(\mathbb {F}_q^n\) 的 t 维子空间和 \(\mathbb {F}_q^n\) 的 t 维子空间的基的普通泛化。我们利用 \({{\,\textrm{GL}\,}}(n,q)\) 的特征理论给出了 \({{\,\textrm{GL}\,}}(n,q)\) 传递子集的结构特征,并将这些子集解释为 \({{\,\textrm{GL}\,}}(n,q)\) 共轭类关联方案中的设计。特别地,我们概括了佩林关于在 t 维子空间上横向作用的 \({{\,\textrm{GL}\,}}(n,q)\) 子群的定理。我们考察了 \({{\,\textrm{GL}\,}}(n,q)\) 的传递子群,发现 \({{\,\textrm{GL}\,}}(n,q)\) 的子群中没有 \(1<;t<n\) 在 t 维子空间上起传递作用,除非它包含 \({{\,\textrm{SL}\,}}(n,q)\) 或者是两个例外群之一。另一方面,对于所有固定的 t,我们证明了存在着 \({{\,\textrm{GL}\,}}(n,q)\) 的非难子集,这些子集在 \(\mathbb {F}_q^n\) 的线性独立 t 元组上是传递的、这也说明了在\({{\,\textrm{GL}\,}}(n,q)\)的非重子集上存在传递性的更一般的类旗结构。我们建立了与({{\,\textrm{GL}\,}}(n,q)\)正交多项式(即 Al-Salam-Carlitz 多项式)的联系,并推广了 Rudvalis 和 Shinoda 关于 \({{\,\textrm{GL}\,}}(n,q)\) 中元素的定点数分布的结果。我们的许多结果可以解释为对称群相应结果的 q-analogs.
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来源期刊
CiteScore
1.60
自引率
0.00%
发文量
236
审稿时长
3-6 weeks
期刊介绍: "Mathematische Zeitschrift" is devoted to pure and applied mathematics. Reviews, problems etc. will not be published.
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