On isometric full embeddings of symplectic dual polar spaces into Hermitian dual polar spaces
type: Item
instance of: scholarly article
title: On isometric full embeddings of symplectic dual polar spaces into Hermitian dual polar spaces
zbMATH Open document ID: 1166.51002
published in: Linear Algebra and its Applications
publication date: 2009-04-29T00:00:00Z
review text: Let \({\mathbb K}'\), \({\mathbb K}\) be (finite or infinite) fields such that \({\mathbb K}'\) is a quadratic Galois extension of \({\mathbb K}\) and let \(\theta\) denote the unique nontrivial element in \(\text{Gal}({\mathbb K}'/{\mathbb K})\). The dual polar space \(DW(2n-1,{\mathbb K})\), \(n\geq\,2\), has up to equivalence a unique isometric full embedding into the Hermitian dual polar space \(DH(2n-1,{\mathbb K}',\theta)\) (the proof of this fact in [\textit{B. De Bruyn}, Finite Fields Appl. 14, No.~1, 188--200 (2008; Zbl 1139.51009)] for the finite case can be extended to the infinite case). The Grassmann-embedding of \(DH(2n-1,{\mathbb K}',\theta)\) [compare \textit{B. De Bruyn}, Linear Multilinear Algebra 56, No.~6, 665--677 (2008; Zbl 1155.51001)] induces a projective embedding of \(DW(2n-1,{\mathbb K})\) which is, as the author shows, isomorphic to the Grassmann-embedding of \(DW(2n-1,{\mathbb K})\). Furthermore, the author proves: If \(n\) is even, then the set of points of \(DH(2n-1,{\mathbb K}',\theta)\) at distance at most \(n/2-1\) from \(DW(2n-1,{\mathbb K})\) is a hyperplane of \(DH(2n-1,{\mathbb K}',\theta)\) which arises from the Grassmann-embedding of \(DH(2n-1,{\mathbb K}',\theta)\).
reviewed by: Rolf Riesinger
Mathematics Subject Classification ID: 51A45
Mathematics Subject Classification ID: 51A50
zbMATH DE Number: 5549298
zbMATH Keywords: symplectic/Hermitian dual polar space
zbMATH Keywords: Hermitian variety
zbMATH Keywords: Grassmann-embedding
zbMATH Keywords: isometric full embedding
zbMATH Keywords: hyperplane arising from an embedding
zbMATH Keywords: geometry of hyperbolic lines of \(W(2n-1,{\mathbb K})\)
author: Bart De Bruyn
MaRDI profile type: Publication
full work available at URL: j.laa.2008.12.025
OpenAlex ID: W2057021224
cites work: Q4385460
cites work: On the generation of dual polar spaces of unitary type over finite fields
cites work: A decomposition of the natural embedding spaces for the symplectic dual polar spaces
cites work: Isometric full embeddings of \(DW(2n - 1,q)\) into \(DH(2n - 1,q^{2}\))
cites work: On the Grassmann-embeddings of the hermitian dual polar spaces
cites work: Some subspaces of the \(k\)th exterior power of a symplectic vector space
cites work: On Veldkamp lines
DOI: 10.1016/J.LAA.2008.12.025
description: scientific article; zbMATH DE number 5549298