When $P$ is the maximal parabolic corresponding to the spin node in type $B_n$, the homogeneous coordinate ring of $G/P$ is generated by the generalized minors associated to the weights in the Weyl group orbit of $\frac{1}{2}(e_1 + \ldots + e_n)$, which correspond to the subsets of $\{1,...,n\}$. I would like to realize this $G/P$ as the space of $n$-dimensional isotropic subspaces of $\mathbb{C}^{2n+1}$. This is because I'm working with a toric chart which comes from taking a Lusztig/Berenstein--Fomin--Zelevinsky-style parametrization of a double Bruhat cell in $SO(2n+1)$ and then "projecting" to the last $n$ rows of the matrix.
I'm confused about how to compute the generalized minors in this realization. I've identified $2^n$ maximal minors which are perfect squares in terms of the parameters of my chart, so the generalized minors ought to be their positive square roots. But how do I know that there are regular functions on $G/P$ which restrict to the positive square roots of the appropriate maximal minors in this chart? And is there a simple way to compute these functions on all of $G/P$?
