A question on Grassmannian

Question

Let $T: Gr(n,\mathbb C^{2n}) \rightarrow G(n,\mathbb C^{2n})$ be the involution defined by $W \rightarrow W^{\perp}$ with respect to a symplectic form on $\mathbb C^{2n}$. Is there a direct proof (without referring to the cohomology) of the fact that $T$ induces an involution on the set of young diagrams contained in rectangular diagram of shape $n \times n$ ? And, the involution takes a young diagram to its transpose.