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 SU 2 n ( F ) ( n ∈ N ) has generating rank ( 2 n k ) when F ≠ F 4 . We also reprove the main result of Blok (2007) [3], namely that the Grassmannian of totally isotropic k-spaces associated to the symplectic group Sp 2 n ( F ) has generating rank ( 2 n k ) − ( 2 n k − 2 ) , when Char ( F ) ≠ 2 .