Three homogeneous embeddings of DW(2n-1,2)

We construct a homogeneous full projective embedding of the dual polar space DW(2n-1,2) from the hyperplane intersections of hyperbolic type of the parabolic quadric Q(2n,2). We believe that this embedding is universal, but have not succeeded in proving that. As a by-product of our investigations, we have obtained necessary and sufficient conditions for this to be the case and came across two other homogeneous full projective embeddings of DW(2n-1,2), one with vector dimension 22n-1+32n-1-23and another one with vector dimension 22n-1+32n-1-2-6n3-2-6n