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On isometric full embeddings of symplectic dual polar spaces into Hermitian dual polar spaces

✍ Scribed by Bart De Bruyn

Publisher: Elsevier Science
Year: 2009
Tongue: English
Weight: 192 KB
Volume: 430
Category: Article
ISSN: 0024-3795
DOI: 10.1016/j.laa.2008.12.025

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✦ Synopsis

Let n 2, let K, K be fields such that K is a quadratic Galoisextension of K and let θ denote the unique nontrivial element in Gal(K /K). Suppose the symplectic dual polar space DW (2n -1, K) is fully and isometrically embedded into the Hermitian dual polar space DH(2n -1, K , θ). We prove that the projective embed- ding of DW (2n -1, K) induced by the Grassmann-embedding of DH(2n -1, K , θ) is isomorphic to the Grassmann-embedding of DW (2n -1, K). We also prove that if n is even, then the set of points of DH(2n -1, K , θ) at distance at most n 2 -1 from DW (2n -1, K) is a hyperplane of DH(2n -1, K , θ) which arises from the Grassmann-embedding of DH(2n -1, K , θ).

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