Toric Degeneration of Schubert Varieties and Gelfand–cetlin Polytopes

This note constructs the flat toric degeneration of the manifold Fln of flags in Cn from [GL96] as an explicit GIT quotient of the Gröbner degeneration in [KM03]. This implies that Schubert varieties degenerate to reduced unions of toric varieties, associated to faces indexed by rc-graphs (reduced pipe dreams) in the Gelfand–Cetlin polytope. Our explicit description of the toric degeneration of Fln provides a simple explanation of how Gel fand–Cetlin decompositions for irreducible polynomial representations of GLn arise via geometric quantization.