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 , 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 $DQ(2n,q)$, $DQ^-(2n+1,q)$, $DH(2n-1,q^2)$ and $DW(2n-1,q)$ ($q$ odd), but the latter only for $n\leq 5$. 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 $DW(2n-1,q)$.

Cardinali, I., DE BRUYN, B., Pasini, A. (2007). Minimal full polarized embeddings of dual polar spaces. JOURNAL OF ALGEBRAIC COMBINATORICS, 25, 7-23 [10.1007/s10801-006-0013-8].

### Minimal full polarized embeddings of dual polar spaces

#### Abstract

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 , 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 $DQ(2n,q)$, $DQ^-(2n+1,q)$, $DH(2n-1,q^2)$ and $DW(2n-1,q)$ ($q$ odd), but the latter only for $n\leq 5$. 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 $DW(2n-1,q)$.
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2007
Cardinali, I., DE BRUYN, B., Pasini, A. (2007). Minimal full polarized embeddings of dual polar spaces. JOURNAL OF ALGEBRAIC COMBINATORICS, 25, 7-23 [10.1007/s10801-006-0013-8].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11365/35380