On extensions of hyperplanes of dual polar spaces

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• 作者：Bart De Bruyn^a ; ^{bdb@cage.ugent.be}
• 关键词：Dual polar space ; Absolutely universal embedding ; Minimal full polarized embedding ; Grassmann embedding ; (Extension of) hyperplanes
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2011
• 出版时间：April 2011
• 年：2011
• 卷：118
• 期：3
• 页码：949-961
• 全文大小：219 K

文摘

Let Δ be a thick dual polar space and F a convex subspace of diameter at least 2 of Δ. Every hyperplane G of the subgeometry of Δ induced on F will give rise to a hyperplane H of Δ, the so-called extension of G. We show that F and G are in some sense uniquely determined by H. We also consider the following problem: if e is a full projective embedding of Δ and if e_F is the full embedding of induced by e, does the fact that G arises from the embedding e_F imply that H arises from the embedding e? We will study this problem in the cases that e is an absolutely universal embedding, a minimal full polarized embedding or a Grassmann embedding of a symplectic dual polar space. Our study will allow us to prove that if e is absolutely universal, then also e_F is absolutely universal.