Assume you are given two subspace $V$ and $W$, belonging to some Schubert cells $C_I$ and $C_J$. Is there an elementary closed form description of $V\cap W$ in $C_{I\cap J}$?
Here by "elementary" I mean everything is with respect to the standard flag, and thus the subspaces can be thought as column space of matrices of appropriate size, and $I, J$ are subsets of row/column indices.
Again, I am not looking for a description that involves, say, algebraic geometry language or other alikes.