The relational complexity of linear groups acting on subspaces

TL;DR

This paper precisely determines the relational complexity of certain linear groups acting on subspaces, extending known results and providing bounds for broader classes of groups and actions.

Contribution

It computes the exact relational complexity for groups between PSL and PGL acting on 1-spaces, and extends bounds to groups between PSL(q) and PΓL(q) on m-spaces.

Findings

Exact relational complexity for groups between PSL and PGL on 1-spaces.

Bounds on relational complexity for groups between PSL(q) and PΓL(q).

Generalization to actions on m-spaces for m ≥ 1.

Abstract

The relational complexity of a subgroup $G$ of $\mathrm{Sym}(\Omega)$ is a measure of the way in which the orbits of $G$ on $\Omega^k$ for various $k$ determine the original action of $G$. Very few precise values of relational complexity are known. This paper determines the exact relational complexity of all groups lying between $\mathrm{PSL}_{n}(\mathbb{F})$ and $\mathrm{PGL}_{n}(\mathbb{F})$, for an arbitrary field $\mathbb{F}$, acting on the set of $1$-dimensional subspaces of $\mathbb{F}^n$. We also bound the relational complexity of all groups lying between $\mathrm{PSL}_{n}(q)$ and $\mathrm{P}\Gamma\mathrm{L}_{n}(q)$, and generalise these results to the action on $m$-spaces for $m \ge 1$.