Algebras and Representation Theory

, Volume 18, Issue 6, pp 1481–1503 | Cite as

Minuscule Schubert Varieties: Poset Polytopes, PBW-Degenerated Demazure Modules, and Kogan Faces

  • Rekha Biswal
  • Ghislain Fourier


We study a family of posets and the associated chain and order polytopes. We identify the order polytope as a maximal Kogan face in a Gelfand-Tsetlin polytope of a multiple of a fundamental weight. We show that the character of such a Kogan face equals to the character of a Demazure module which occurs in the irreducible representation of \(\mathfrak {sl}_{n+1}\) having highest weight multiple of fundamental weight and for any such Demazure module there exists a corresponding poset and associated maximal Kogan face. We prove that the chain polytope parametrizes a monomial basis of the associated PBW-graded Demazure module and further, that the Demazure module is a favourable module, e.g. interesting geometric properties are governed by combinatorics of convex polytopes. Thus, we obtain for any minuscule Schubert variety a flat degeneration into a toric projective variety which is projectively normal and arithmetically Cohen-Macaulay. We provide a necessary and sufficient condition on the Weyl group element such that the toric variety associated to the chain polytope and the toric variety associated to the order polytope are isomorphic.


Poset polytopes PBW filtration Schubert varieties Kogan faces 

Mathematics Subject Classification (2010)

05E10 14M25 14M1 17B10 


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  1. 1.
    Alexeev, V., Brion, M.: Toric degenerations of spherical varieties. Selecta Math. (N.S.) 10(4), 453–478 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Ardila, F., Bliem, T., Salazar, D.: Gelfand-Tsetlin polytopes and Feigin-Fourier-Littelmann-Vinberg polytopes as marked poset polytopes. J. Combin. Theory Ser. A 118(8), 2454–2462 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Backhaus, T., Bossinger, L., Desczyk, C., Fourier, G.: The degree of the Hilbert-Poincaré polynomial of PBW-graded modules. C. R. Math. Acad. Sci. Paris 352(12), 959–963 (2014)Google Scholar
  4. 4.
    Backhaus, T., Desczyk, C.: PBW filtration: Feigin-Fourier-Littelmann modules via Hasse diagrams. arXiv:1407.73664. J. Lie Theory 25(3), 815–856 (2015)
  5. 5.
    Cherednik, I., Feigin, E.: Extremal part of the PBW-filtration and E-polynomials. arXiv:1306.3146 (2013)
  6. 6.
    Cherednik, I., Orr, D.: Nonsymmetric difference Whittaker functions. arXiv:1302.4094 (2013)
  7. 7.
    De Loera, J.A., McAllister, T.B.: Vertices of Gelfand-Tsetlin polytopes. Discrete Comput. Geom. 32(4), 459–470 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Feigin, E., Fourier, G., Littelmann, P.: PBW filtration and bases for irreducible modules in type A n. Transform. Groups 16(1), 71–89 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Feigin, E., Fourier, G., Littelmann, P.: PBW filtration and bases for symplectic Lie algebras. Int. Math. Res. Not. IMRN 1(24), 5760–5784 (2011)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Feigin, E., Fourier, G., Littelmann, P.: Favourable modules: Filtrations, polytopes, Newton-Okounkov bodies and flat degenerations. arXiv:1306.1292v3 (2013)
  11. 11.
    Feigin, E., Fourier, G., Littelmann, P.: PBW-filtration over \(\mathbb {Z}\) and compatible bases for v(λ) in type A n and C n. Springer Proceedings in Mathematics and Statistics 40, 35–63 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Feigin, E., Makedonskyi, I.: Nonsymmetric Macdonald polynomials, Demazure modules and PBW filtration. Preprint arXiv:1407.6316 (2014)
  13. 13.
    Fourier, G.: New homogeneous ideals for current algebras: Filtrations, fusion products and Pieri rules, Preprint: arXiv:1403.4758. Moscow M. Journ. 15(1), 49–72 (2015)
  14. 14.
    Fourier, G.: PBW-degenerated Demazure modules and Schubert varieties for triangular elements. arXiv:1408.6939 (2014)
  15. 15.
    Gelfand, I.M., Cetlin, M.L.: Finite-dimensional representations of the group of unimodular matrices. Doklady Akad. Nauk SSSR (N.S.) 71, 825–828 (1950)MathSciNetGoogle Scholar
  16. 16.
    Gonciulea, N., Lakshmibai, V.: Degenerations of flag and Schubert varieties to toric varieties. Transform. Groups 1(3), 215–248 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Hibi, T., Li, N.: Unimodular equivalence of order and chain polytopes. arXiv:1208.4029 (2012)
  18. 18.
    Kashiwara, M.: The crystal base and Littelmann’s refined Demazure character formula. Duke Math. J. 71(3), 839–858 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Kogan, M.: Schubert geometry of flag varieties and Gelfand-Cetlin theory. PhD-thesis (2000)Google Scholar
  20. 20.
    Kirichenko, V.A., Smirnov, E.Yu., Timorin, V.A.: Schubert calculus and Gelfand-Tsetlin polytopes. Uspekhi Mat. Nauk. 67(4(406)), 89–128 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Littelmann, P.: Crystal graphs and Young tableaux. J. Algebra 175(1), 65–87 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Littelmann, P.: Cones, crystals, and patterns. Transform. Groups 3(2), 145–179 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Stanley, R.P.: Hilbert functions of graded algebras. Adv. Math. 28(1), 57–83 (1978)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Stanley, R.P.: Two poset polytopes. Discrete Comput. Geom. 1(1), 9–23 (1986)CrossRefMathSciNetzbMATHGoogle Scholar

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© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.The Institute of Mathematical SciencesCIT CampusTaramaniIndia
  2. 2.Mathematisches InstitutUniversität BonnBonnGermany
  3. 3.School of Mathematics and StatisticsUniversity of GlasgowGlasgowUK

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