Computing Toric Degenerations of Flag Varieties

Part of the Fields Institute Communications book series (FIC, volume 80)


We compute toric degenerations arising from the tropicalization of the full flag varieties \(\mathop{\mathrm{Fl}}\nolimits _{4}\) and \(\mathop{\mathrm{Fl}}\nolimits _{5}\) embedded in a product of Grassmannians. For \(\mathop{\mathrm{Fl}}\nolimits _{4}\) and \(\mathop{\mathrm{Fl}}\nolimits _{5}\) we compare toric degenerations arising from string polytopes and the FFLV polytope with those obtained from the tropicalization of the flag varieties. We also present a general procedure to find toric degenerations in the cases where the initial ideal arising from a cone of the tropicalization of a variety is not prime.

MSC 2010 codes:

14T05 13P10 14M25 14M15 17B10 



This article was initiated during the Apprenticeship Weeks (22 August–2 September 2016), led by Bernd Sturmfels, as part of the Combinatorial Algebraic Geometry Semester at the Fields Institute. The authors are grateful to the Max Planck Institute MiS Leipzig, where part of this project was carried out. We are grateful to Diane Maclagan, Kiumars Kaveh, and Kristin Shaw for inspiring conversations. We also would like to thank Diane Maclagan, Yue Ren and five anonymous referees for their comments on an earlier version of this manuscript. Further, L.B. and F.M. would like to thank Ghislain Fourier and Xin Fang for many inspiring discussions. K.M would like to express her gratitude to Dániel Joó for many helpful conversations. F.M. was supported by a postdoctoral fellowship from the Einstein Foundation Berlin. S.L. was supported by EPSRC grant 1499803.


  1. 1.
    Valery Alexeev and Michel Brion: Toric degenerations of spherical varieties, Selecta Math. (N.S.) 10 (2005) 453–478.Google Scholar
  2. 2.
    Dave Anderson: Okounkov bodies and toric degenerations, Math. Ann. 356 (2013) 1183–1202.Google Scholar
  3. 3.
    Arkady Berenstein and Andrei Zelevinsky: Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math. 143 (2001) 77–128.Google Scholar
  4. 4.
    René Birkner: Polyhedra, a package for computations with convex polyhedral objects, J. Softw. Algebra Geom. 1 (2009) 11–15.Google Scholar
  5. 5.
    Lara Bossinger, Xin Fang, Fourier Ghislain, Milena Hering, and Martina Lanini: Toric degenerations of \(\mathop{\mathrm{Gr}}\nolimits (2,n)\) and \(\mathop{\mathrm{Gr}}\nolimits (3,6)\) via plabic graphs, arXiv:1612.03838 [math.CO]Google Scholar
  6. 6.
    Lara Bossinger and Ghislain Fourier: String cone and superpotential combinatorics for flag and schubert varieties in type A, arXiv:1611.06504 [math.RT].Google Scholar
  7. 7.
    Philippe Caldero: Toric degenerations of Schubert varieties, Transform. Groups 7 (2002) 51–60.Google Scholar
  8. 8.
    David Cox, John Little, and Donal O’Shea: Ideals, varieties, and algorithms, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992.Google Scholar
  9. 9.
    David A. Cox, John B. Little, and Henry K. Schenck: Toric varieties, Graduate Studies in Mathematics 124. American Mathematical Society, Providence, RI, 2011.Google Scholar
  10. 10.
    Wolfram Decker, Gert-Martin Greuel, Gerhard Pfister, and Hans Schönemann: Singular 4-1-0, a computer algebra system for polynomial computations, available at
  11. 11.
    David Eisenbud: Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics 150, Springer-Verlag, New York, 1995.Google Scholar
  12. 12.
    Xin Fang, Ghislain Fourier, and Peter Littelmann: Essential bases and toric degenerations arising from birational sequences, Adv. Math. 312 (2017) 107–149.Google Scholar
  13. 13.
    Xin Fang, Ghislain Fourier, and Peter Littelmann: On toric degenerations of flag varieties, Representation theory-current trends and perspectives, 187–232, EMS Series of Congress Reports. European Mathematical Society, Zürich, 2017.Google Scholar
  14. 14.
    Xin Fang, Ghislain Fourier, and Markus Reineke: PBW-type filtration on quantum groups of type A n, J. Algebra 449 (2016) 321–345.Google Scholar
  15. 15.
    Evgeny Feigin, Ghislain Fourier, and Peter Littelmann: PBW filtration and bases for irreducible modules in type A n, Transform. Groups 16 (2011) 71–89.Google Scholar
  16. 16.
    Evgeny Feigin, Ghislain Fourier, and Peter Littelmann: Favourable modules: filtrations, polytopes, Newton–Okounkov bodies and flat degenerations, Transform. Groups 22(2) (2017) 321–352.Google Scholar
  17. 17.
    Ewgenij Gawrilow and Michael Joswig: polymake: a framework for analyzing convex polytopes. In Polytopes-combinatorics and computation (Oberwolfach, 1997), 43–73, DMV Sem. 29, Birkhäuser, Basel, 2000.Google Scholar
  18. 18.
    Oleg Gleizer and Alexander Postnikov: Littlewood-Richardson coefficients via Yang-Baxter equation, Internat. Math. Res. Notices (2000) 741–774.Google Scholar
  19. 19.
    Daniel R. Grayson and Michael E. Stillman: Macaulay2, a software system for research in algebraic geometry, available at
  20. 20.
    James E. Humphreys: Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics 9. Springer-Verlag, New York-Berlin, 1972.Google Scholar
  21. 21.
    Anders N. Jensen: Gfan, a software system for Gröbner fans and tropical varieties, available at
  22. 22.
    Anders N. Jensen: Gfan version 0.5: A users manual, available at
  23. 23.
    Kiumars Kaveh and Christopher Manon: Khovanskii bases, higher rank valuations and tropical geometry, arXiv:1610.00298 [math.AG].Google Scholar
  24. 24.
    Peter Littelmann: Cones, crystals, and patterns, Transform. Groups 3 (1998) 145–179.Google Scholar
  25. 25.
    Diane Maclagan and Bernd Sturmfels: Introduction to Tropical Geometry, Graduate Studies in Mathematics 161, American Mathematical Society, RI, 2015.Google Scholar
  26. 26.
    Ezra Miller and Bernd Sturmfels: Combinatorial Commutative Algebra, Graduate Texts in Mathematics 227, Springer-Verlag, New York, 2005.Google Scholar
  27. 27.
    Fatemeh Mohammadi and Kristin Shaw: Toric degenerations of Grassmannians from matching fields. In preparation, 2016.Google Scholar
  28. 28.
    Gerhard Pfister, Wolfram Decker, Hans Schoenemann, and Santiago Laplagne: Primdec.lib, a Singular library for computing the primary decomposition and radical of ideals.Google Scholar
  29. 29.
    David Speyer and Bernd Sturmfels: The tropical Grassmannian, Adv. Geom. 4 (2004) 389–411.Google Scholar
  30. 30.
    Bernd Sturmfels: Fitness, Apprenticeship, and Polynomials, in Combinatorial Algebraic Geometry, 1–19, Fields Inst. Commun. 80, Fields Inst. Res. Math. Sci., 2017.Google Scholar
  31. 31.
    Bernd Sturmfels: Gröbner bases and convex polytopes, University Lecture Series 8, American Mathematical Society, Providence, RI, 1996.Google Scholar
  32. 32.
    Jakub Witaszek: The degeneration of the Grassmannian into a toric variety and the calculation of the eigenspaces of a torus action, J. Algebr. Stat. 6 (2015) 62–79.Google Scholar

Copyright information

© Springer Science+Business Media LLC 2017

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversity of CologneCologneGermany
  2. 2.Mathematics InstituteUniversity of WarwickCoventryUK
  3. 3.Department of MathematicsYale UniversityNew HavenUSA
  4. 4.School of MathematicsUniversity of Bristol, University WalkBristolUK

Personalised recommendations