Gelfand-Zetlin polytopes and flag varieties

Details Gelfand-Zetlin polytopes and flag varieties Gelfand-Zetlin polytopes and flag varieties

2009-06-26

arxiv.org/abs/0906.4866v1

International Mathematics Research Notices (2009), 20 pages, doi:10.1093/imrn/rnp223

Mathematics

Algebraic Geometry

16 pages, 2 figures

Scientific paper

10.1093/imrn/rnp223

I construct a correspondence between the Schubert cycles on the variety of complete flags in C^n and some faces of the Gelfand-Zetlin polytope associated with the irreducible representation of SL_n(C) with a strictly dominant highest weight. The construction is based on a geometric presentation of Schubert cells by Bernstein-Gelfand-Gelfand using Demazure modules. The correspondence between the Schubert cycles and faces is then used to interpret the classical Chevalley formula in Schubert calculus in terms of the Gelfand-Zetlin polytopes. The whole picture resembles the picture for toric varieties and their polytopes.

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