Isotropic Subspaces of Schur Modules

It is a well-known fact that over the complex numbers and for a fixed $k$ and $n$, a generic $s$ in $Sym^2V^*$ vanishes on some $k$-dimensional subspace of $V$ if and only if $n\geq 2k$. Tevelev found exact conditions for the extension of this statement for general symmetric and skew-symmetric multilinear forms, and we extend his work to all possible symmetric types, which corresponds to Schur modules for a general partition.