The generating rank of a polar Grassmannian

Abstract: In this paper we compute the generating rank of k-polar Grassmannians defined over commutative division rings. Among the new results, we compute the generating rank of k-Grassmannians arising from Hermitian forms of Witt index n defined over vector spaces of dimension N>2n. We also study generating sets for the 2-Grassmannians arising from quadratic forms of Witt index n defined over V(N,Fq) for q=4,8,9 and 2n≤N≤2n+2. We prove that for N>6 they can be generated over the prime subfield, thus determining their generating rank.