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Direct Constructions of Hyperplanes of Dual Polar Spaces Arising from Embeddings
Bart De Bruyn
Department of Mathematics, Ghent University, Krijgslaan 281 (S22), B-9000 Gent, Belgium
bdb@cage.ugent.be
Annals of Combinatorics 14 (2) pp.193-209 Summer, 2010
AMS Subject Classification: 51A45, 51A50, 51E20
Abstract:
Let e be one of the following full projective embeddings of a finite dual polar space Δ of rank n≥2: (i) The Grassmann-embedding of the symplectic dual polar space Δ
DW(2n–1, q); (ii) the Grassmann-embedding of the Hermitian dual polar space Δ DH(2n–1, q2); (iii) the spin-embedding of the orthogonal dual polar space Δ DQ(2n, q);
(iv) the spin-embedding of the orthogonal dual polar space Δ DQ-(2n+1, q). Let He denote the set of all hyperplanes of D arising from the embedding e. We give a method for constructing
the hyperplanes of He without implementing the embedding e and discuss (possible) applications of the given construction.
Keywords: dual polar space, hyperplane, Grassmann-embedding, spin-embedding

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