Transitivity in finite general linear groups

TL;DR

This paper studies transitive subsets of the general linear group over finite fields, characterizing their structure, classifying subgroups with certain transitivity properties, and connecting these findings to algebraic combinatorics and orthogonal polynomials.

Contribution

It provides structural characterizations of transitive subsets of (n,q) using character theory, generalizes known theorems, and explores the existence of nontrivial transitive subsets on flag-like structures.

Findings

Classifies subgroups of (n,q) acting transitively on t-dimensional subspaces.

Shows existence of nontrivial subsets transitive on linearly independent t-tuples.

Connects transitivity properties with orthogonal polynomials and q-analogs.

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 $\operatorname{GL}(n,q)$ 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.