global section of some sheaves

Question

Let $\mathrm{Grass}(r,V)$ be the Grassmannian over a field $k$. What is $H^0(\Sigma^{\alpha}(S))$, where $S$ is the tautological sheaf and $\Sigma$ the Schur functor. In characteristic zero this is isomorphic to $\Sigma^{\alpha}(V^*)$, for $|\alpha|>0$. Does this hold in arbitrary characteristic?