Consider the Lie group $G=SL_n(\mathbb C)$ with Borel subgroup $B$ and maximal torus $T\subset B$. I'm interested in the (Zariski) closures of $T$-orbits in the flag variety $F=G/B$.
Now, as far as I can tell, for a generic point in $F$ this closure is the toric variety associated with the permutahedron. Further, let us choose an element $w$ of the Weyl group and let $X_w\subset F$ be the corresponding Schubert variety. Then for a generic point in $X_w$ the closure seems to be the toric variety associated with the convex hull of vertices of the permutahedron corresponding to $w'\ge w$ with respect to the Bruhat order, i.e. the convex hull of the weight diagram of the corresponding Demazure module in an irrep with a regular highest weight.
1) Is this last description accurate?
2) My main question. Are, in fact, all orbits of this form? More precisely, let $S$ be the set of all points in $F$ that are generic in some Schubert variety in the above sense. Is it then true that any point outside of $S$ can be mapped to a point in $S$ by the action of the Weyl group?
References to literature are much appreciated.