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The generating rank of the unitary and symplectic Grassmannians

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 ${2n\choose k}$ 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 ${2n\choose k}-{2n\choose k-2}$, when $\rm{Char}(\mathbb{F})\ne 2$.