The proportion of non-degenerate complementary subspaces in classical spaces

TL;DR

This paper establishes a lower bound on the proportion of pairs of subspaces with trivial intersection in finite classical spaces, aiding in group recognition algorithms, especially for orthogonal groups over fields of order 2.

Contribution

It provides a new lower bound for the proportion of non-degenerate complementary subspaces intersecting trivially, including cases previously unresolved.

Findings

Lower bound depends on subset sizes, dimensions, and field size

Bound applies to classical spaces, including orthogonal groups over 

Advances understanding of subspace interactions in finite classical geometries

Abstract

Given positive integers $e_1,e_2$, let $X_i$ denote the set of $e_i$-dimensional subspaces of a fixed finite vector space $V=(\mathbb{F}_q)^{e_1+e_2}$. Let $Y_i$ be a non-empty subset of $X_i$ and let $\alpha_i=|Y_i|/|X_i|$. We give a positive lower bound, depending only on $\alpha_1,\alpha_2,e_1,e_2,q$, for the proportion of pairs $(S_1,S_2)\in Y_1\times Y_2$ which intersect trivially. As an application, we bound the proportion of pairs of non-degenerate subspaces of complementary dimensions in a finite classical space that intersect trivially. This problem is motivated by an algorithm for recognizing classical groups. By using techniques from algebraic graph theory, we are able to handle orthogonal groups over the field of order 2, a case which had eluded Niemeyer, Praeger, and the first author.