The probability of spanning a classical space by two non-degenerate subspaces of complementary dimension

2021/09/21 by S. P. Glasby, Alice C. Niemeyer, Glasby, S. P. +3
Mathematics · #05-08 #15A63 #20P05 #Advanced Algebra and Geometry #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2109.10015

openalex publication_date 2021/09/21 · openalex created_date 2021/09/27 · openalex updated_date 2026/07/28

Abstract

Let n,n' be positive integers and let V be an (n+n')-dimensional vector space over a finite field \mathbbF equipped with a non-degenerate alternating, hermitian or quadratic form. We estimate the proportion of pairs (U, U'), where U is a non-degenerate n-subspace and U' is a non-degenerate n'-subspace of V, such that U+ U'=V (usually such spaces U and U' are not perpendicular). The proportion is shown to be at least 1-c/|\mathbbF| for some constant c\leqslant 2 in the symplectic or unitary cases, and c<3 in the orthogonal case.

Related

• The proportion of non-degenerate complementary subspaces in classical spaces
• Random generation of direct sums of finite non-degenerate subspaces
• Anzahl theorems for disjoint subspaces generating a non-degenerate subspace II: quadratic forms
• An application of Bertini Theorem
• Orthogonal systems in vector spaces over finite fields