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

2003/03/17 by M. G. Kogan, Mikhail Kogan, Ezra Miller +2 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #math.AG

paper · pdf · doi:10.48550/arxiv.math/0303208

14 pages

arxiv created 2003/03/17 · openalex publication_date 2003/03/17 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This note constructs the flat toric degeneration of the manifold FL_n of flags in Cⁿ 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.

Citations

• Flags, Schubert polynomials, degeneracy loci, and determinantal formulas
• A geometric Littlewood-Richardson rule

Cited by

• Structure and geometry of the tableaux algebra

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