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Uspekhi Mat. Nauk, 2012, Volume 67, Issue 4(406), Pages 89–128 (Mi umn9492)  

This article is cited in 12 scientific papers (total in 12 papers)

Schubert calculus and Gelfand–Zetlin polytopes

V. A. Kirichenkoab, E. Yu. Smirnovcb, V. A. Timorindb

a Institute for Information Transmission Problems of the Russian Academy of Sciences, Moscow, Russia
b National Research University Higher School of Economics
c Laboratoire J.-V. Poncelet (UMI 2615 du CNRS)
d Independent University of Moscow

Abstract: A new approach is described to the Schubert calculus on complete flag varieties, using the volume polynomial associated with Gelfand–Zetlin polytopes. This approach makes it possible to compute the intersection products of Schubert cycles by intersecting faces of a polytope.
Bibliography: 23 titles.

Keywords: Flag variety, Schubert calculus, Gelfand–Zetlin polytope, volume polynomial.

DOI: https://doi.org/10.4213/rm9492

Full text: PDF file (873 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2012, 67:4, 685–719

Bibliographic databases:

Document Type: Article
UDC: 512.734
MSC: Primary 14L30; Secondary 52B20, 14M15, 14N15
Received: 25.05.2012

Citation: V. A. Kirichenko, E. Yu. Smirnov, V. A. Timorin, “Schubert calculus and Gelfand–Zetlin polytopes”, Uspekhi Mat. Nauk, 67:4(406) (2012), 89–128; Russian Math. Surveys, 67:4 (2012), 685–719

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. P. Gusev, V. Kiritchenko, V. Timorin, “Counting vertices in Gelfand–Zetlin polytopes”, J. Comb. Theory, Ser. A, 120:4 (2013), 960–969  crossref  mathscinet  zmath  isi  elib  scopus
    2. N. Perrin, “On the geometry of spherical varieties”, Transform. Groups, 19:1 (2014), 171–223  crossref  mathscinet  zmath  isi  elib  scopus
    3. R. Biswal, G. Fourier, “Minuscule Schubert varieties: poset polytopes, PBW-degenerated Demazure modules, and Kogan faces”, Algebr. Represent. Theory, 18:6 (2015), 1481–1503  crossref  mathscinet  zmath  isi  scopus
    4. K. Kaveh, “Crystal bases and Newton-Okounkov bodies”, Duke Math. J., 164:13 (2015), 2461–2506  crossref  mathscinet  zmath  isi  scopus
    5. G. Fourier, “PBW-degenerated Demazure modules and Schubert varieties for triangular elements”, J. Combin. Theory Ser. A, 139 (2016), 132–152  crossref  mathscinet  zmath  isi  scopus
    6. G. Fourier, “Marked poset polytopes: Minkowski sums, indecomposables, and unimodular equivalence”, J. Pure Appl. Algebra, 220:2 (2016), 606–620  crossref  mathscinet  zmath  isi  scopus
    7. I. Yu. Makhlin, “Brion's Theorem for Gelfand–Tsetlin Polytopes”, Funct. Anal. Appl., 50:2 (2016), 98–106  mathnet  crossref  crossref  mathscinet  isi  elib
    8. V. Kiritchenko, “Geometric mitosis”, Math. Res. Lett., 23:4 (2016), 1071–1097  crossref  mathscinet  zmath  isi  elib  scopus
    9. R. A. Proctor, M. J. Willis, “Parabolic Catalan numbers count flagged Schur functions and their appearances as type A Demazure characters (key polynomials)”, Discret. Math. Theor. Comput. Sci., 19:3 (2017)  mathscinet  isi
    10. X. Fang, G. Fourier, P. Littelmann, “On toric degenerations of flag varieties”, Representation theory—current trends and perspectives, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2017, 187–232  mathscinet  zmath  isi
    11. B. H. An, Yu. Cho, J.S. Kim, “On the $f$-vectors of Gelfand-Cetlin polytopes”, European J. Combin., 67 (2018), 61–77  crossref  mathscinet  zmath  isi  scopus
    12. Proctor R.A., Willis M.J., “Convexity of Tableau Sets For Type a Demazure Characters (Key Polynomials), Parabolic Catalan Numbers”, Discret. Math. Theor. Comput. Sci., 20:2 (2018), UNSP 3  isi
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