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

TL;DR

This work analyzes the bipartite $q$-Kneser graphs $ ext{Gamma}_{e_1,e_2}$ associated to $e_i$-subspaces of a $d$-dimensional space over $\mathbb{F}_q$, establishing a precise link between graph-theoretic invariants and algebraic generation questions in $GL_d(q)$. It derives an explicit formula for the proportion $P(e_1,e_2)$ of closed 3-arcs among 3-walks and proves that, for groups with $SL_d(q)\le G\le GL_d(q)$, the proportion of irreducible $(e_1,e_2)$-stingray duos coincides with $P(e_1,e_2)$; it also provides tight upper and lower bounds, notably $1-q^{-1}-q^{-2}<P(e_1,e_2)<1-q^{-1}-q^{-2}+2q^{-3}-2q^{-5}$ for $2\le e_2\le e_1$ and $q\ge2$. The authors connect these combinatorial counts to irreducibility and generation in linear groups, and show how these results improve complexity analyses for classical-group recognition algorithms, particularly in small $q$. Overall, the paper blends elementary linear-algebraic counting with group-theoretic structure to yield precise probabilistic estimates with practical algorithmic implications.

Abstract

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' $Γ_{e_1,e_2}$ 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.

Figures (1)

• Figure 1: 3-walks $(F_1,U_1,U_2,F_2)$ with possible repeated vertices; in the second, third and fourth diagrams we have $U_1=F_2$, $U_2=F_1$ and $(U_1,U_2)=(F_2,F_1)$, respectively

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 2.1
• Lemma 2.2
• proof
• Remark 2.3
• Definition 3.1
• Definition 3.2
• Remark 3.3