Three homogeneous embeddings of $$\textit{DW}(2n-1,2)$$ DW ( 2 n - 1 , 2 )

Abstract

We construct a homogeneous full projective embedding of the dual polar space $$\textit{DW}(2n-1,2)$$ 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 $$\textit{DW}(2n-1,2)$$ , one with vector dimension $$\frac{2^{2n-1} + 3 \cdot 2^{n-1} -2}{3}$$ and another one with vector dimension $$\frac{2^{2n-1} + 3 \cdot 2^{n-1} -2 - 6n}{3}$$ .