Minimal full polarized embeddings of dual polar spaces (Q855307)

The authors study thick dual polar spaces \(\Delta\) of rank \(n \geq 2\) which admit a full polarized embedding \(e : \Delta \to \Sigma\) into a finite-dimensional projective space \(\Sigma\). For any point \(x\) of \( \Delta\) such an embedding \(e\) maps the set \(H_{x}\) of points of \( \Delta\) at non-maximal distance from \(x\) (with respect to the collinearity graph of \(\Delta\)) into some hyperplane \(e^\ast(x) := e (H_{x}) \) of \(\Sigma\). Based on a result of \textit{A. Kasikova} and \textit{E. Shult} [J. Algebra 238, No. 1, 265--291 (2001; Zbl 0988.51001)] it is shown that up to isomorphism there exists a unique full polarized embedding of \(\Delta\) of minimal dimension. It turns out that this embedding is equivalent to the dual embedding \(e^\ast : \Delta \to \Sigma^\ast\) of \(e\) into the dual space \(\Sigma^\ast\) of \( \Sigma\). In the last chapter the minimal full polarized embeddings of the finite dual polar spaces \(DQ(2n,q)\), \(DQ^-(2n+1,q)\), \(DH(2n-1,q^2) \) for \(q\) odd and of \(DW(2n-1,q)\) for \(q\leq 5\) odd are determined.