Symplectic subspaces of symplectic Grassmannians (Q881813)

The author proves some properties of symplectic Grassmannians and classifies subspaces \(S\) of \(C_{q}(F)\) where \(S\cong C_{m,k}(F)\) when \(\text{char}(F)\neq 2\). Besides, the author shows that if \(\text{char}(F)\neq 2\) and \(S\) be a subspace of \(C_{n,q}(F)\), \(S\cong C_{m,k}(F)\); then there exists a totally isotropic subspace \(A\) of dimension \(q-k\) and a subspace \(B\) such that \(A\subset B\subset A^{\perp }\), \(B/A\) non-degenerate and \(S=T_{q}(B,A)\).