I'm looking for convenient coordinates to use to describe the intersection of a Schubert cell $X^\circ_\lambda$ and an opposite Schubert cell $\Omega^\circ_\mu$ in a Grassmannian $G(k,n)$.
Describing a single Schubert variety is no problem as its maximal cell is isomorphic to an affine space $\mathbb{A^r}$ -- explicitly written down as matrices with many entries set to 0 or 1, and the rest free. And there's the corresponding description for opposite Schuberts. For describing the intersection $X^\circ_\lambda \cap \Omega^\circ_\mu$, is there a simple way to 'combine' the forward- and opposite-Schubert descriptions? For example, in $G(2,5)$,
$$\begin{pmatrix} * &* & 1 & 0 & 0 \\ * & * & 0 & * & 1 \end{pmatrix} \cap \begin{pmatrix} 1 & * & 0 & * & *\\ 0 & 0 & 1 & * & * \end{pmatrix}$$ has a dense open subset consisting of matrices of the form $$\begin{pmatrix} *' & * & 1 & 0 & 0 \\ 0 & 0 & *' & * & 1 \end{pmatrix},$$ where $*'$ means the entry can't be zero. (Note that we can't just "overlay" the two types of matrix, since e.g. they disagree in the middle column.)
Ideally I'd like to have exactly $\dim(X^\circ_\lambda \cap \Omega^\circ_\mu)$ coordinates, as in this example. (And, while I'm happy to describe, e.g. a dense open subset of $X^\circ_\lambda \cap \Omega^\circ_\mu$, it would be helpful to know what exactly I'm ignoring.)