Random generation of direct sums of finite non-degenerate subspaces

Abstract

Let $V$ be a $d$-dimensional vector space over a finite field $\mathbb{F}$ equipped with a non-degenerate hermitian, alternating, or quadratic form. Suppose $|\mathbb{F}|=q^2$ if $V$ is hermitian, and $|\mathbb{F}|=q$ otherwise. Given integers $e,e^{\prime }$ such that $e+e^{\prime }⩽d$, we estimate the proportion of pairs $(U,U^{\prime })$, where $U$ is a non-degenerate $e$-subspace of $V$ and $U^{\prime }$ is a non-degenerate $e^{\prime }$-subspace of $V$, such that $U\cap U^{\prime }=0$ and $U\oplus U^{\prime }$ is non-degenerate (the sum $U\oplus U^{\prime }$ is direct and usually not perpendicular). The proportion is shown to be positive and at least $1-c/q>0$ for some constant $c$. For example, $c=7/4$ suffices in both the unitary and symplectic cases. The arguments in the orthogonal case are delicate and assume that $\dim (U)$ and $\dim (U^{\prime })$ are even, an assumption relevant for an algorithmic application (which we discuss) for recognising finite classical groups. We also describe how recognising a classical groups $G$ relies on a connection between certain pairs $(U,U^{\prime })$ of non-degenerate subspaces and certain pairs $(g,g^{\prime })\in G^2$ of group elements where $U=\mathrm{im}(g-1)$ and $U^{\prime }=\mathrm{im}(g^{\prime }-1)$.

24 pages, 2 tables, minor changes to Def 3.6(a,b). Proof of Lemma 3.7(a) and Prop 4.11 slightly shortened