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 (2nk) 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 (2nk)-(2nk-2), when Char(F) 2.

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