Toric Schubert varieties and directed Dynkin diagrams

A flag variety is a homogenous variety $G/B$ where $G$ is a simple algebraic group over the complex numbers and $B$ is a Boel subgroup of $G$. A Schubert variety $X_w$ is a subvariety of $G/B$ indexed by an element $w$ in the Weyl group of $G$. It is called toric if it is a toric variety with respect to the maximal torus of $G$ in $B$. In this paper, we associate an edge-labeled digraph $\mathcal{G}_w$ with a toric Schubert variety $X_w$ and classify toric Schubert varieties up to isomorphism. We also give a simple criterion of when a toric Schubert variety $X_w$ is (weak) Fano in terms of $\mathcal{G}_w$. Finally, we discuss whether toric Schubert varieties can be distinguished by their integral cohomology rings up to isomorphism and show that this is the case when $G$ is of simply-laced type.