Grassmannians, flag varieties, and Gelfand-Zetlin polytopes
Evgeny Smirnov
TL;DR
This paper introduces Schubert calculus on Grassmannians and flag varieties, exploring cohomology, polynomials, and a new combinatorial approach using Gelfand-Zetlin polytopes, with applications to enumerative geometry.
Contribution
It presents a survey of Schubert calculus and introduces a novel combinatorial method for full flag varieties via Gelfand-Zetlin polytopes.
Findings
Analysis of cohomology ring structures
Connections between Schur and Schubert polynomials
New combinatorial approach to Schubert calculus
Abstract
These are extended notes of a talk given at Maurice Auslander Distinguished Lectures and International Conference (Woods Hole, MA) in April 2013. Their aim is to give an introduction into Schubert calculus on Grassmannians and flag varieties. We discuss various aspects of Schubert calculus, such as applications to enumerative geometry, structure of the cohomology rings of Grassmannians and flag varieties, Schur and Schubert polynomials. We conclude with a survey of results of V.Kiritchenko, V.Timorin and the author on a new approach to Schubert calculus on full flag varieties via combinatorics of Gelfand-Zetlin polytopes.
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