Random generation of direct sums of finite non-degenerate subspaces

TL;DR

This paper estimates the probability of randomly chosen pairs of non-degenerate subspaces in a finite classical vector space forming a direct sum, with implications for recognizing classical groups.

Contribution

It provides bounds on the proportion of such subspace pairs, including detailed analysis for orthogonal cases, aiding algorithms for classical group recognition.

Findings

Proportion of suitable subspace pairs is at least 1 - c/q, with c=7/4 in key cases.

The analysis applies to hermitian, symplectic, and orthogonal forms.

Connections between subspace pairs and group elements facilitate group recognition algorithms.

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 $e, e'$ such that $e+e'\leqslant d$, we estimate the proportion of pairs $(U, U')$, where $U$ is a non-degenerate $e$-subspace of $V$ and $U'$ is a non-degenerate $e'$-subspace of $V$, such that $U\cap U'=0$ and $U\oplus U'$ is non-degenerate (the sum $U\oplus U'$ 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 $\dim(U')$ 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 $(U,U')$ of non-degenerate subspaces and certain pairs $(g,g')\in G^2$ of group elements where $U={\rm im}(g-1)$ and $U'={\rm im}(g'-1)$.