Direct Constructions of Hyperplanes of Dual Polar Spaces Arising from Embeddings

Abstract

Let e be one of the following full projective embeddings of a finite dual polar space ∆ of rank n ≥ 2: (i) the Grassmann-embedding of the symplectic dual polar space ∆ ∼ = DW (2n − 1, q); (ii) the Grassmannembedding of the Hermitian dual polar space ∆ ∼ = DH(2n − 1, q 2 ); (iii) the spin-embedding of the orthogonal dual polar space ∆ ∼ = DQ(2n, q); (iv) the spin-embedding of the orthogonal dual polar space ∆ ∼ = DQ − (2n + 1, q). Let H e denote the set of all hyperplanes of ∆ arising from the embedding e. We give a method for constructing the hyperplanes of H e without implementing the embedding e and discuss (possible) applications of the given construction.