The relational complexity of linear groups acting on subspaces (Q6611927)

Let \(H\) be a group acting on a set \(\Omega \). The relational complexity \( \mathrm{RC}(H,\Omega )\) is defined to be the least integer \(k>0\) such that for each \(h\geq k\), the \(H\)-orbit of each \(h\)-tuple in \(\Omega ^{h}\) is determined by the orbits of its \(k\)-subtuples. This concept originates in model theory (see, for example, [G. Cherlin, On the relational complexity of a finite permutation group, J. Algebraic Combin. 43 (2016), no. 2, 339--374].) Exact values of \(\mathrm{RC}(H,\Omega )\) are only known for a few cases.\N\NIn the present paper, the authors consider \(\mathrm{RC}(\bar{H},\Omega _{m})\) where \(\bar{H}\) is an almost simple group with \(\mathrm{PSL}_{n}(\mathbb{F})\leq \bar{ H}\leq \) \(\mathrm{P}\Gamma \mathrm{L}_{n}(\mathbb{F})\) (for an arbitrary, possibly infinite field \(\mathbb{F}\)) and \(\Omega _{m}\) is the projective space of all subspaces of dimension \(m\) in \(\mathbb{F}^{n}\). Define \(\omega (k)\) to be the number of distinct prime divisors of the integer \(k\). The following are the main results. (Theorem A): If \(n\geq 3\) then: (i) \(\mathrm{RC}(\mathrm{PGL}_{n}( \mathbb{F}),\Omega _{1}=n\) if \(\vert \mathbb{F}\vert \leq 3\) and \( n+2\) otherwise; if \(\bar{H}<\mathrm{PGL}_{n}(\mathbb{F)}\) then \(\mathrm{RC}(\bar{H} ,\Omega _{1})=2n-1\) if \(n=3\) and \(2n-2\) otherwise. \N\NTheorem B. Suppose that \(\mathbb{F=F}_{q}\) and \(e=:\vert \bar{H}:\bar{H}\cap \mathrm{PGL}_{n}(\mathbb{F} _{q})\vert >1\), then: \N\begin{enumerate}\N\item for \(n=2\) and \(q\geq 8\) we have \(4+\omega (e)\) \(\geq \mathrm{RC}(\bar{H},\Omega _{1})\geq 4\) except when \(\bar{H} =\mathrm{P}\Sigma \mathrm{L}_{2}(9)\); \N\item for \(n=3\) we have \(2n-1+\omega (e)\geq \mathrm{ RC}(\bar{H},\Omega _{1})\geq n+2\).\end{enumerate} \N\NTheorem C. Under the same hypotheses as Theorem B, we have \(2+\omega (e)\) \(\geq \) \(\mathrm{RC}(\bar{H},\Omega _{m})-mn+m^{2}\) \(\geq 1\) for all \(m\) satisfying \(2\leq m\leq n/2\).\N\NThere are more precise results in special cases.