Closures of torus orbits in flag varieties 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.
Questions.
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.
 A: A point in the Grassmannian $ x \in G(k, n) $ defines a matroid $ M = M(x)$.  Associated to this matroid is a matroid polytope $P(M)$.  The torus orbit closure through $ x $ is the toric variety associated to $ P(M) $.
Similarly, if we take a point $ x $ in the flag variety, then the can associate a flag matroid $M(x) $ and a flag matroid polytope $ P(M)$. Again the torus orbit closure through $ x $ is the toric variety associated to $ P(M) $.
Thus the classification of torus orbit closures in the flag variety is given by realizable flag matroids. So the answer your second question is no. 

The theory of flag matroids (in arbitrary type) and their polytopes was developed in the excellent classic paper by GGMS:
I. M. Gelfand, R. M. Goresky, R. D. MacPherson and
V. V. Serganova, Combinatorial Geometries, Convex Polyhedra,
and Schubert cells, Adv. Math., 63 (1987), 301–316.
Alex Yong's lecture notes are also quite helpful:
https://faculty.math.illinois.edu/~ayong/Math595TheGrassmannian/Grlecture2matroids.pdf
