Skip to main content
SpringerLink
Log in

Computing Toric Degenerations of Flag Varieties

  • Chapter
  • First Online:
Combinatorial Algebraic Geometry

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

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Valery Alexeev and Michel Brion: Toric degenerations of spherical varieties, Selecta Math. (N.S.) 10 (2005) 453–478.

    Google Scholar 

  2. Dave Anderson: Okounkov bodies and toric degenerations, Math. Ann. 356 (2013) 1183–1202.

    Google Scholar 

  3. Arkady Berenstein and Andrei Zelevinsky: Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math. 143 (2001) 77–128.

    Google Scholar 

  4. René Birkner: Polyhedra, a package for computations with convex polyhedral objects, J. Softw. Algebra Geom. 1 (2009) 11–15.

    Google Scholar 

  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. 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. Philippe Caldero: Toric degenerations of Schubert varieties, Transform. Groups 7 (2002) 51–60.

    Google Scholar 

  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. 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. Wolfram Decker, Gert-Martin Greuel, Gerhard Pfister, and Hans Schönemann: Singular 4-1-0, a computer algebra system for polynomial computations, available at www.singular.uni-kl.de.

  11. David Eisenbud: Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics 150, Springer-Verlag, New York, 1995.

    Google Scholar 

  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. 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. 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. 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. 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. 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. Oleg Gleizer and Alexander Postnikov: Littlewood-Richardson coefficients via Yang-Baxter equation, Internat. Math. Res. Notices (2000) 741–774.

    Google Scholar 

  19. Daniel R. Grayson and Michael E. Stillman: Macaulay2, a software system for research in algebraic geometry, available at www.math.uiuc.edu/Macaulay2/.

  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. Anders N. Jensen: Gfan, a software system for Gröbner fans and tropical varieties, available at home.imf.au.dk/jensen/software/gfan/gfan.html.

  22. Anders N. Jensen: Gfan version 0.5: A users manual, available at home.math.au.dk/jensen/software/gfan/gfanmanual0.5.pdf.

  23. Kiumars Kaveh and Christopher Manon: Khovanskii bases, higher rank valuations and tropical geometry, arXiv:1610.00298 [math.AG].

    Google Scholar 

  24. Peter Littelmann: Cones, crystals, and patterns, Transform. Groups 3 (1998) 145–179.

    Google Scholar 

  25. Diane Maclagan and Bernd Sturmfels: Introduction to Tropical Geometry, Graduate Studies in Mathematics 161, American Mathematical Society, RI, 2015.

    Google Scholar 

  26. Ezra Miller and Bernd Sturmfels: Combinatorial Commutative Algebra, Graduate Texts in Mathematics 227, Springer-Verlag, New York, 2005.

    Google Scholar 

  27. Fatemeh Mohammadi and Kristin Shaw: Toric degenerations of Grassmannians from matching fields. In preparation, 2016.

    Google Scholar 

  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. David Speyer and Bernd Sturmfels: The tropical Grassmannian, Adv. Geom. 4 (2004) 389–411.

    Google Scholar 

  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. Bernd Sturmfels: Gröbner bases and convex polytopes, University Lecture Series 8, American Mathematical Society, Providence, RI, 1996.

    Google Scholar 

  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 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Mathematisches Institut, University of Cologne, Weyertal 86-90, 50931, Cologne, Germany

    Lara Bossinger

  2. Mathematics Institute, University of Warwick, Zeeman Building, Coventry, CV4 7AL, UK

    Sara Lamboglia

  3. Department of Mathematics, Yale University, 10 Hillhouse Ave., New Haven, CT, 06511, USA

    Kalina Mincheva

  4. School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, UK

    Fatemeh Mohammadi

Authors
  1. Lara Bossinger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sara Lamboglia
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kalina Mincheva
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Fatemeh Mohammadi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fatemeh Mohammadi .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada

    Gregory G. Smith

  2. Department of Mathematics, University of California, Berkeley, California, USA

    Bernd Sturmfels

Appendix

Appendix

In this Appendix, we provide numerical evidence of our computations. Algebraic and Combinatorial Invariants of \(\mathop{\mathrm{trop}}\nolimits (\mathop{\mathrm{Fl}}\nolimits _{5})\) Table 4 contains data on the non-prime maximal cones of \(\mathop{\mathrm{trop}}\nolimits (\mathop{\mathrm{Fl}}\nolimits _{5})\).

In Table 5, there is information on the polytopes obtained from maximal prime cones of \(\mathop{\mathrm{trop}}\nolimits (\mathop{\mathrm{Fl}}\nolimits _{5})\). It shows the f-vectors of the polytopes associated to maximal prime cones of \(\mathop{\mathrm{trop}}\nolimits (\mathop{\mathrm{Fl}}\nolimits _{5})\) for one representative in each orbit. The last column contains information on the existence of a combinatorial equivalence between these polytopes and the string polytopes resp. FFLV polytope for ρ. The initial ideals are all Cohen–Macaulay .

Table 4 Data for non-prime initial ideals of \(\mathop{\mathrm{Fl}}\nolimits _{5}\)
Full size table
Table 5 Information on the polytopes obtained from maximal prime cones of \(\mathop{\mathrm{trop}}\nolimits (\mathop{\mathrm{Fl}}\nolimits _{5})\)
Full size table

Algebraic Invariants of the \(\mathop{\mathrm{Fl}}\nolimits _{5}\) String Polytopes Table 6 contains information on the string polytopes and FFLV polytope for \(\mathop{\mathrm{Fl}}\nolimits _{5}\), such as the weight vectors constructed in Sect. 5, primeness of the initial ideals with respect to these vectors, and the MP property. The last column contains information on unimodular equivalences among these polytopes. If there is no information in this column, then there is no unimodular equivalence between this polytope and any other polytope in the table.

Table 6 Polytopes for \(\mathop{\mathrm{Fl}}\nolimits _{5}\)
Full size table

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer Science+Business Media LLC

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Bossinger, L., Lamboglia, S., Mincheva, K., Mohammadi, F. (2017). Computing Toric Degenerations of Flag Varieties. In: Smith, G., Sturmfels, B. (eds) Combinatorial Algebraic Geometry. Fields Institute Communications, vol 80. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-7486-3_12

Download citation

Publish with us

Policies and ethics

Search

Navigation