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Home > Vol 25, No 1 (2007)

Ilaria Cardinali¹ , Bart De Bruyn² and Antonio Pasini¹

¹Università di Siena Dipartimento di Scienze Matematiche e Informatiche `R. Magari' Pian dei Mantellini, 44 I-53100 Siena Italy Pian dei Mantellini, 44 I-53100 Siena Italy
²Ghent University Department of Pure Mathematics and Computer Algebra Galglaan, 2 B-9000 Gent Belgium Galglaan, 2 B-9000 Gent Belgium

Let Δ  be a thick dual polar space of rank n \geq  2 admitting a full polarized embedding e in a finite-dimensional projective space Σ , i.e., for every point x of Δ , e maps the set of points of Δ  at non-maximal distance from x into a hyperplane e\ast ( x) of Σ . 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 Δ  of minimal dimension. We also show that e\ast  realizes a full polarized embedding of Δ  into a subspace of the dual of Σ , and that e\ast  is isomorphic to the minimal full polarized embedding of Δ . In the final section, we will determine the minimal full polarized embeddings of the finite dual polar spaces DQ(2 n, q), DQ  ^- (2 n+1, q), DH(2 n - 1, q ²) and DW(2 n - 1, q) ( q odd), but the latter only for n\leq  5. We shall prove that the minimal full polarized embeddings of DQ(2 n, q), DQ  ^- (2 n+1, q) and DH(2 n - 1, q ²) are the `natural' ones, whereas this is not always the case for DW(2 n - 1, q).

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