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 $V(N,{\mathbb{F}}_q)$ for $q=4,8,9$ and $2n\le N\le 2n+2$. We prove that for $N>6$ they can be generated over the prime subfield, thus determining their generating rank.

32 pages