Irredundant bases for finite groups of Lie type (Q6043861)

Let \(G\) be a finite group acting on a set \(\Omega \) and for any list \( \Lambda :=[\omega _{1},\omega _{2},\dots,\omega _{t}]\) of points from \(\Omega \) let \(G_{(\Lambda )}=G_{\omega_{1},\omega _{2},\dots,\omega_{t}}\) be the pointwise stabilizer of \(\Lambda \). Then \(\Lambda \) is a base for \(G\ \)if \( G_{(\Lambda)}=1\) and \(\Lambda \) is an irredundant base if no proper sublist is a base. Let \(b(G,\Omega )\) denote the length of the shortest base for \( (G,\Omega )\) and let \(I(G,\Omega )\) denote the length of the longest possible irredundant base. A theorem proved by \textit{M. W. Liebeck} and \textit{A. Shalev} [J. Am. Math. Soc. 12, No. 2, 497--520 (1999; Zbl 0916.20003)] (conjectured by Cameron and Kantor) is that \(b(G,\Omega )\) is universally bounded over the set of almost simple primitive nonstandard permutation groups. The main theorem of the present paper is: If \(G\) is a finite simple group of Lie type of rank \(r\) acting primitively on \(\Omega \), then \(I(G,\Omega )\leq 174r^{8}\). In this estimate, the constant \(174\) is not sharp and the exponent \(8\) is probably far from sharp but examples show it must be at least \(2\). The lower bound on \(I(G,\Omega )\) is trivial; for example, if we take \(G:=\mathrm{SL}_{r}(2)\) with \(r\) an odd prime and \(\Omega \) the set of cosets of the normalizer of a Singer cycle, then \(I(G,\Omega )\leq 3\).