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

Mikhail Kogan·Ezra Miller
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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.

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Mikhail Kogan, Ezra Miller. 2003-03-17. Toric degeneration of Schubert varieties and Gel'fand-Cetlin polytopes. https://arxiv.org/abs/math/0303208

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