On the Grassmann-embeddings of the hermitian dual polar spaces

Abstract

Let H denote a hermitian variety of Witt-index n ≥ 2 in PG(2n–1, 𝕂) and let θ denote the associated involutory automorphism of 𝕂. The dual polar space Δ associated with H has a full projective embedding e (the so-called Grassmann-embedding) into PG(W), where W is a -dimensional vector space over the fix-field 𝕂0 of θ. The projective space PG(W) can be regarded as a Baer-subgeometry of PG(∧ n V), where V is a 2n-dimensional vector space over 𝕂, equipped with a hermitian form defining H. In this article, we determine the precise equations for the Baer-subgeometry PG(W) of PG(∧ n V) and give a proof for the fact that e is an embedding without the use of any group-theoretical considerations. Subsequently, we will prove a decomposition theorem for the embedding e in the same spirit as the decomposition theorem for the Grassmann-embeddings of the symplectic dual polar spaces [B. De Bruyn, A decomposition of the natural embedding spaces for the symplectic dual polar spaces. Linear Algebra and its Applications, in press].