A few days ago I asked the following question at MSE and received no answer. I thought I would try here.

Let $\xi$ be an integral dominant weight of an irreducible root system $\Delta$, and let $\mathcal{O}_{\xi}$ be its orbit under the action of the Weyl group. The elements of the orbit are the vertices of an n-dimensional convex polytope.

Is there an efficient way to pick out all the subsets of points in $\mathcal{O}_{\xi}$ forming the faces of the convex polytope? I would like to have an algorithm that takes a Weyl orbit of an integral dominant weight as input and gives as output a list of subsets of vertices forming the faces of the polytope. By efficient, I mean that I would like something better than a brute force algorithm.

Let $G$ be the compact connected Lie group corresponding to the root system $\Delta$, and let $T\subset G$ be a maximal torus. We could also view the orbit $\mathcal{O}_{\xi}$ as the moment map image for a Hamiltonian group action of $T$ on $G/T$. In this case, the faces of the moment map image can be viewed as the image of points fixed under a one parameter subgroup of $T$.

Is there a way to understand which 1-parameter subgroups have fixed point sets that get mapped to the faces of the moment map image?