Toric degeneration of Schubert varieties and Gelfand–Tsetlin polytopes

Abstract

This note constructs the flat toric degeneration of the manifold Fℓ n of flags in C n due to Gonciulea and Lakshmibai (Transform. Groups 1(3) (1996) 215) as an explicit GIT quotient of the Gröbner degeneration due to Knutson and Miller (Gröbner geometry of Schubert polynomials, Ann. Math. (2) to appear). 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–Tsetlin polytope. Our explicit description of the toric degeneration of Fℓ n provides a simple explanation of how Gelfand–Tsetlin decompositions for irreducible polynomial representations of GL n arise via geometric quantization.