Bipartite q-Kneser graphs and two-generated irreducible linear groups

Abstract

Let $V:={\left(F_q\right)}^d$ be a d-dimensional vector space over the field $F_q$ 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 ${\mathrm{SL}}_d\left(q\right)⊴G⩽{\mathrm{GL}}_d\left(q\right)$, 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 $2⩽e_2⩽e_1$ and $q⩾2$. 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.