Toric Schubert varieties and directed Dynkin diagrams

A flag variety is a smooth projective homogeneous variety G / B , where G is a simple algebraic group over the complex numbers and B is a Borel subgroup of G . A Schubert variety X w is a subvariety of G / B indexed by an element w of the Weyl group of G . It is called toric if it is a toric variety with respect to the action of the maximal torus T ⊂ B . In this paper, we associate an edge-labeled digraph 𝒢 w to a toric Schubert variety X w and classify toric Schubert varieties up to isomorphism. We also give a simple criterion for when a toric Schubert variety X w is (weak) Fano in terms of 𝒢 w . Finally, we discuss whether toric Schubert varieties can be distinguished by their integral cohomology rings up to isomorphism and show that this holds when G is simply-laced.