The generating rank of the unitary and symplectic Grassmannians

Abstract

We prove that the Grassmannian of totally isotropic $k$-spaces of the polar space associated to the unitary group ${\mathsf{SU}}_{2n}(\mathbb{F})$ ($n\in \mathbb{N}$) has generating rank $\left(\genfrac{}{}{0ex}{}{2n}{k}\right)$ when $\mathbb{F}\ne {\mathbb{F}}_4$. We also reprove the main result of Blok [Blok2007], namely that the Grassmannian of totally isotropic $k$-spaces associated to the symplectic group ${\mathsf{Sp}}_{2n}(\mathbb{F})$ has generating rank $\left(\genfrac{}{}{0ex}{}{2n}{k}\right)-\left(\genfrac{}{}{0ex}{}{2n}{k-2}\right)$, when $\mathrm{Char}(\mathbb{F})\ne 2$.