Minimal full polarized embeddings of dual polar spaces

Cardinali, Ilaria ; De Bruyn, Bart ; Pasini, Antonio

Journal of Algebraic Combinatorics, Tome 25 (2007) no. 1, pp. 7-23.

Résumé

Summary: Let $\Delta $be a thick dual polar space of rank $n \geq 2$ admitting a full polarized embedding $e$ in a finite-dimensional projective space $\Sigma $, i.e., for every point $x$ of $\Delta , e$ maps the set of points of $\Delta $at non-maximal distance from $x$ into a hyperplane $e\ast ( x)$ of $\Sigma $. Using a result of Kasikova and Shult [11], we are able the show that there exists up to isomorphisms a unique full polarized embedding of $\Delta $of minimal dimension. We also show that $e\ast $realizes a full polarized embedding of $\Delta $into a subspace of the dual of $\Sigma $, and that $e\ast $is isomorphic to the minimal full polarized embedding of $\Delta $. In the final section, we will determine the minimal full polarized embeddings of the finite dual polar spaces $D$Q$(2 n, q), DQ ^{ - }(2 n+1, q), D$H($2 n - 1, q ^{2}$) and $D$W$(2 n - 1, q) ( q$ odd), but the latter only for $n\leq 5$. We shall prove that the minimal full polarized embeddings of $D$Q$(2 n, q), DQ ^{ - }(2 n+1, q)$ and $D$H($2 n - 1, q ^{2}$) are the `natural' ones, whereas this is not always the case for $D$W($2 n - 1, q$).

Détail

Keywords: keywords dual polar space, polarized embedding, universal embedding