Bipartite \(q\)-Kneser graphs and two-generated irreducible linear groups

• Published in: arXiv.org
• Main Authors: Glasby, S P, Niemeyer, Alice C, Praeger, Cheryl E
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 09.12.2023
• Subjects: Algorithms; Fields (mathematics); Graph theory; Graphical representations; Group theory; Subgroups; Vector spaces
• ISSN: 2331-8422

Let \(V:=(\mathbb{F}_q)^d\) be a \(d\)-dimensional vector space over the field \(\mathbb{F}_q\) of order \(q\). Fix positive integers \(e_1,e_2\) satisfying \(e_1+e_2=d\). Motivated by analysing a fundamental algorithm in computational group theory for recognising classical groups, we consider a certain quantity \(P(e_1,e_2)\) which arises in both graph theory and group representation theory: \(P(e_1,e_2)\) is the proportion of \(3\)-walks in the `bipartite \(q\)-Kneser graph' \(\Gamma_{e_1,e_2}\) that are closed \(3\)-arcs. We prove that, for a group \(G\) satisfying \({\rm SL}_d(q)\leqslant G\leqslant{\rm GL}_d(q)\), the proportion of certain element-pairs in \(G\) called `\((e_1,e_2)\)-stingray duos' which generate an irreducible subgroup is also equal to \(P(e_1,e_2)\). We give an exact formula for \(P(e_1,e_2)\), and prove that \(1-q^{-1}-q^{-2}< P(e_1,e_2)< 1-q^{-1}-q^{-2}+2q^{-3}-2q^{-5}\) for \(2\leqslant e_2\leqslant e_1\) and \(q\geqslant2\).These bounds have implications for the complexity analysis of the state-of-the-art algorithms to recognise classical groups, which we discuss in the final section.