Toric variety defined by the Weyl orbit of a minuscule weight

Question

Let $\Phi$ be a (reduced, crystallographic) root system with Weyl group $\mathcal{W}$, and $p$ a (nonzero) minuscule weight for $\Phi$: its orbit $\mathcal{W}p$ is the set of vertices of a convex polytope which, given together with the affine lattice $p+L$ generated by these vertices (a translate of the root lattice inside the weight lattice), defines a projective toric variety $X$.

Question: Is there a more geometric description of this toric variety $X$?

Specifically, can we relate it to the partial flag variety $G/P$ quotient of the semisimple algebraic group $G$ associated to $\Phi$ by the parabolic subgroup associated to $p$? Can we ($T$-equivariently? canonically?) embed $X$ in $G/P$?

Note: A presumably closely related toric variety is the one associated with the fan of Weyl chambers and the weight lattice: see here for a discussion and the paper by de Mari, Procesi & Shayman, "Hessenberg Varieties" (Trans. Amer. Math. Soc. 332 (1992), 529–534); this is the closure of a general orbit of a maximal torus $T \subset G$ acting on the flag variety $G/B$: see Batyrev & Blume, "The Functor of Toric Varieties Associated with Weyl Chambers", Tohoku Math. J. 63 (2011), 581–604 and the refernces to Klyachko found there. But to be honest, while I write "closely related" because intuitively it should be, I don't really see the details of this close relation.

Answer 1

In general if $T$ acts on a projective variety $X$ with moment polytope $\Phi(X)$, then a general point $x\in X$ will have $\Phi(\overline{T\cdot x}) = \Phi(X)$ i.e. be an abnormal toric variety with this same polytope. In the case of Grassmannians, Fink and Speyer prove that the variety is in fact normal, and I expect this works in other minuscule cases.

The close relation you claim is a map: take $y\in G/B$, $\pi:G/B\to G/P$, and look at $\overline{T\cdot y} \twoheadrightarrow \overline{T\cdot \pi(y)}$, inducing maps between their moment polytopes and normalizations.