Toric degeneration of Schubert varieties and Gel'fand-Cetlin polytopes

TL;DR

This paper constructs an explicit toric degeneration of flag manifolds and Schubert varieties, linking geometric quantization with Gel'fand-Cetlin polytopes, and explains their combinatorial structure via GIT quotients.

Contribution

It provides an explicit GIT quotient construction of the toric degeneration of flag varieties and Schubert varieties, connecting geometric and combinatorial aspects.

Findings

Schubert varieties degenerate to unions of toric varieties

Gel'fand-Cetlin decompositions arise from geometric quantization

Explicit description of the toric degeneration of FL_n

Abstract

This note constructs the flat toric degeneration of the manifold FL_n of flags in C^n from [Gonciulea-Lakshmibai 96] as an explicit GIT quotient of the Gr"obner degeneration in [Knutson-Miller 03]. This implies that Schubert varieties degenerate to reduced unions of toric varieties, associated to faces indexed by rc-graphs (reduced pipe dreams) in the Gel'fand-Cetlin polytope. Our explicit description of the toric degeneration of FL_n provides a simple explanation of how Gel'fand-Cetlin decompositions for irreducible polynomial representations of GL_n arise via geometric quantization.