A note on embedding and generating dual polar spaces (Q2725174)

A set \(X \not=\emptyset\) of points of a point-line geometry \({\mathcal S}\) is called a generating set if it is not contained in a proper subspace of \({\mathcal S }\). The generating rank is the smallest size of a generating set. NEWLINENEWLINENEWLINEIn the paper generating sets of cardinality \(2^n\) are constructed for the unitary dual polar space \(DU(2n+1,q^2)\) and the elliptic othogonal dual polar space \(DO^-(2n+2,q)\). It is shown that the automorphism group of each geometry acts transitively on those subgraphs of the collinearity graph that are \(n\)-hypercubes. Such an \(n\)-hypercube is proven to generate the whole geometry. It is also shown that the elliptic dual polar space \(DO^+(2n+2,q)\) has an absolutely universal embedding in \text{ PG}\((2^n-1,q^2)\). NEWLINENEWLINENEWLINEAlso a survey of the current knowledge on generating ranks and universal embedding spaces of dual polar spaces is included.