Minimal full polarized embeddings of dual polar spaces |
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Authors: | Ilaria Cardinali Bart De Bruyn Antonio Pasini |
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Affiliation: | (1) Dipartimento di Scienze Matematiche e Informatiche ‘R. Magari’, Università di Siena, Pian dei Mantellini, 44, I-53100 Siena, Italy;(2) Department of Pure Mathematics and Computer Algebra, Ghent University, Galglaan, 2, B-9000 Gent, Belgium |
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Abstract: | Let Δ be a thick dual polar space of rank n ≥ 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∗(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∗ realizes a full polarized embedding of Δ into a subspace of the dual of Σ, and that e∗ 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(2n,q), DQ −(2n+1,q), DH(2n−1,q 2) and DW(2n−1,q) (q odd), but the latter only for n≤ 5. We shall prove that the minimal full polarized embeddings of DQ(2n,q), DQ −(2n+1,q) and DH(2n−1,q 2) are the `natural' ones, whereas this is not always the case for DW(2n−1, q).B. De Bruyn: Postdoctoral Fellow of the Research Foundation - Flanders. |
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Keywords: | Dual polar space Polarized embedding Universal embedding |
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