Derivation of \(m\)-systems
type: Item
instance of: scholarly article
title: Derivation of \(m\)-systems
zbMATH Open document ID: 1020.51003
DOI: 10.1016/S0195-6698(03)00011-8
published in: European Journal of Combinatorics
publication date: 2003-04-02T00:00:00Z
review text: A partial \(m\)-system \(\mathcal M\) of a polar space \(\mathcal P\) in \(\text{PG}(n, q)\) is a set of \(m\)-dimensional totally singular subspaces \(\pi_1,\dots,\pi_r\) of \(\mathcal P\) such that each generator of \(\mathcal P\) containing an element \(\pi_i\in\mathcal M\) has an empty intersection with \((\pi_1\cup\dots\cup\pi_r)\setminus\pi_i\). Partial \(m\)-systems of polar spaces were introduced by \textit{E. E. Shult} and \textit{J. A. Thas} [J. Comb. Theory, Ser. A 68, 184-204 (1994; Zbl 0824.51004)]. They show that there exists an upper bound, which is independent of \(m\), on the number of elements of a partial \(m\)-system and they call a partial \(m\)-system which meets this upper bound an \(m\)-system. The authors discuss a way to construct new \(m\)-systems from known ones by replacing a set of subspaces of the \(m\)-system by a suitably chosen other set of subspaces, such that the unions of the elements for both sets coincide. This technique is called derivation of \(m\)-systems and it is described in the following theorem: Let \(\mathcal M :=\{\pi_1,\pi_2,\dots,\pi_r\}\) be an \(m\)-system of a finite classical polar space \(\mathcal P_n\) in \(\text{PG}(n, q)\). Suppose that there exists a \(\text{PG}(2m+1, q)\) containing \(1\leqslant s \leqslant r\) elements \(\pi_1,\pi_2,\dots,\pi_r\) of \(\mathcal M\). If in \(\text{PG}(2m+q, q)\cap\mathcal P_n\) there exist \(s\) disjoint \(m\)-dimensional totally singular subspaces \(\alpha_1,\alpha_2,\dots,\alpha_s\) of \(\mathcal P_n\) such that the point sets \(\pi_1\cup\pi_2\cup\dots\cup\pi_s\) and \(\alpha_1\cup\alpha_2\cup\dots\cup\alpha_s\) coincide, then \(\mathcal M' := \{\alpha_1,\alpha_2,\dots,\alpha_s,\pi_{s+1}, \pi_{s+2},\dots,\pi_r\}\) is also an \(m\)-system of \(\mathcal P_n\). It is clear that this theorem can be applied to any locally Hermitian 1-system \(\mathcal M'\) of a non-singular quadric, in which case a regulus can be replaced by its opposite regulus to obtain another 1-system \(\m…
Mathematics Subject Classification ID: 51E12
Mathematics Subject Classification ID: 51E14
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 51A50
zbMATH DE Number: 1891293
zbMATH Keywords: \(m\)-systems of polar spaces
zbMATH Keywords: \(m\)-dimensional totally singular subspaces
zbMATH Keywords: derivation of \(m\)-systems
zbMATH Keywords: locally Hermitian 1-system
zbMATH Keywords: non-singular quadric
zbMATH Keywords: regulus
zbMATH Keywords: opposite regulus
zbMATH Keywords: spreads
zbMATH Keywords: semipartial geometries
zbMATH Keywords: symplectic polar space
author: Deirdre Luyckx
author: Joseph A. Thas
reviewed by: Serguey M. Pokas
MaRDI profile type: Publication
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full work available at URL: s0195-6698(03)00011-8
OpenAlex ID: W2078597838
description: scientific article; zbMATH DE number 1891293