On the generation of dual polar spaces of symplectic type over finite fields
Let \(\Gamma=(P,L)\) be a point-line geometry. We define the generating rank, \(\text{gr}(\Gamma)\) to be \(\min\{| X|\mid X\subset P,\langle X\rangle_{\Gamma}=P\}\). By a projective embedding of \(\Gamma\) we mean an injective mapping \(e:P\to PG(V)\), \(V\) a vector space over some division ring, such that (i) the space spanned by \(e(P)\) is all of \(PG(V)\) and (ii) for \(l\in L\), \(e(l)\) is a full line of \(PG(V)\). We define the embedding rank, \(\text{er}(\Gamma)\) to be the maximal dimension of a vector space \(V\) for which there exist an embedding into \(PG(V)\). An embedding \(\widehat{e}\) is said to be a universal relative to \(e\) if there is a morphism \(\widehat{\alpha}:\widehat{e} \to e\) such that for any other morphism \(\gamma:\widehat{e} \to e\), \(\widehat{\alpha}\) factors through \(\gamma\). An embedding \(e:P\to PG(W)\) is relatively universal, if it is universal relative to itself. Let \(V\) be a vector space of dimension \(2n\) over \(F_q\) and let \(f:V\times V \to F_q\) be a non-degenerate alternative form. We denote the isotropic subspaces of dimension \(l\) by \(P(l,q)\). Dual polar space \(DSp_{2n}(q)=({\mathcal P},{\mathcal L})\) is the point-line geometry with \({\mathcal P}=P(n,q)\) and \({\mathcal L}=\{l(A)=\{M\in {\mathcal P} | \;A\subset M\} | \;A\in P(n-1,q)\}\). In this paper it is proved that for \(q>2\) the generating rank and embedding rank of \(DSp_{2n}(q)\) is \({2n\choose n}-{2n\choose n-2}\). Moreover, embedding into \(PG(M)\) with \(dim(M)={2n\choose n}-{2n\choose n-2}\) is relatively universal.
scientific article; zbMATH DE number 1212979
scientific article