SpringerLink
Log in

The Volume Polynomial of Regular Semisimple Hessenberg Varieties and the Gelfand—Zetlin Polytope

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

Regular semisimple Hessenberg varieties are subvarieties of the flag variety Flag(ℂn) arising naturally at the intersection of geometry, representation theory, and combinatorics. Recent results of Abe, Horiguchi, Masuda, Murai, and Sato as well as of Abe, DeDieu, Galetto, and Harada relate the volume polynomials of regular semisimple Hessenberg varieties to the volume polynomial of the Gelfand–Zetlin polytope GZ(λ) for λ = (λ1, λ2, …, λn). In the main results of this paper we use and generalize tools developed by Anderson and Tymoczko, by Kiritchenko, Smirnov, and Timorin, and by Postnikov in order to derive an explicit formula for the volume polynomials of regular semisimple Hessenberg varieties in terms of the volumes of certain faces of the Gelfand–Zetlin polytope, and also exhibit a manifestly positive, combinatorial formula for their coefficients with respect to the basis of monomials in the αi:= λi − λi+1. In addition, motivated by these considerations, we carefully analyze the special case of the permutohedral variety, which is also known as the toric variety associated to Weyl chambers. In this case, we obtain an explicit decomposition of the permutohedron (the moment map image of the permutohedral variety) into combinatorial (n − 1)-cubes, and also give a geometric interpretation of this decomposition by expressing the cohomology class of the permutohedral variety in Flag(ℂn) as a sum of the cohomology classes of a certain set of Richardson varieties.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. Abe, L. DeDieu, F. Galetto, and M. Harada, “Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies,” Sel. Math., New Ser. 24(3), 2129–2163 (2018).

    Article  MathSciNet  Google Scholar 

  2. T. Abe, T. Horiguchi, M. Masuda, S. Murai, and T. Sato, “Hessenberg varieties and hyperplane arrangements,” J. Reine Angew. Math., doi: https://doi.org/10.1515/crelle-2018-0039 (2019); ar**v: 1611.00269 [math.AG].

  3. D. Anderson and J. Tymoczko, “Schubert polynomials and classes of Hessenberg varieties,” J. Algebra 323(10), 2605–2623 (2010).

    Article  MathSciNet  Google Scholar 

  4. A. Ayzenberg and M. Masuda, “Volume polynomials and duality algebras of multi-fans,” Arnold Math. J. 2(3), 329–381 (2016).

    Article  MathSciNet  Google Scholar 

  5. M. Brion, “Lectures on the geometry of flag varieties,” in Topics in Cohomological Studies of Algebraic Varieties: Impanga Lecture Notes (Birkhäuser, Basel, 2005), Trends Math., pp. 33–85.

    Chapter  Google Scholar 

  6. P. Brosnan and T. Y. Chow, “Unit interval orders and the dot action on the cohomology of regular semisimple Hessenberg varieties,” Adv. Math. 329, 955–1001 (2018).

    Article  MathSciNet  Google Scholar 

  7. F. De Mari, C. Procesi, and M. A. Shayman, “Hessenberg varieties,” Trans. Am. Math. Soc. 332(2), 529–534 (1992).

    Article  MathSciNet  Google Scholar 

  8. W. Fulton, Young Tableaux. With Applications to Representation Theory and Geometry (Cambridge Univ. Press, Cambridge, 1997), London Math. Soc. Stud. Texts 35.

    MATH  Google Scholar 

  9. M. Guay-Paquet, “A second proof of the Shareshian–Wachs conjecture, by way of a new Hopf algebra,” ar**v: 1601.05498 [math.CO].

  10. M. Harada and M. Precup, “The cohomology of abelian Hessenberg varieties and the Stanley–Stembridge conjecture,” J. Algebr. Comb. (in press); ar**v: 1709.06736 [math.CO].

  11. S. Kaji, “Maple script for equivariant cohomology of flag manifolds,” https://github.com/shizuo-kaji/Maple-flag-cohomology

  12. K. Kaveh, “Crystal bases and Newton–Okounkov bodies,” Duke Math. J. 164(13), 2461–2506 (2015).

    Article  MathSciNet  Google Scholar 

  13. K. Kaveh and A. Khovanskii, “Newton–Okounkov bodies, semigroups of integral points, graded algebras and intersection theory,” Ann. Math., Ser. 2, 176(2), 925–978 (2012).

    Article  MathSciNet  Google Scholar 

  14. V. Kiritchenko, “Gelfand–Zetlin polytopes and flag varieties,” Int. Math. Res. Not. 2010(13), 2512–2531 (2010).

    MathSciNet  MATH  Google Scholar 

  15. V. A. Kirichenko, E. Yu. Smirnov, and V. A. Timorin, “Schubert calculus and Gelfand–Zetlin polytopes,” Russ. Math. Surv. 67(4), 685–719 (2012) [transl. from Usp. Mat. Nauk 67 (4), 89–128 (2012)].

    Article  Google Scholar 

  16. M. Kogan, “Schubert geometry of flag varieties and Gelfand–Cetlin theory,” PhD Thesis (Mass. Inst. Technol., Cambridge, 2000).

    Google Scholar 

  17. E. Lee, M. Masuda, and S. Park, “Toric Bruhat interval polytopes,” ar**v: 1904.10187 [math.CO].

  18. M. Masuda and T. E. Panov, “Semifree circle actions, Bott towers and quasitoric manifolds,” Sb. Math. 199(8), 1201–1223 (2008) [transl. from Mat. Sb. 199 (8), 95–122 (2008)].

    Article  MathSciNet  Google Scholar 

  19. D. Monk, “The geometry of flag manifolds,” Proc. London Math. Soc., Ser. 3, 9, 253–286 (1959).

    Article  MathSciNet  Google Scholar 

  20. A. Postnikov, “Permutohedra, associahedra, and beyond,” Int. Math. Res. Not. 2009(6), 1026–1106 (2009).

    Article  MathSciNet  Google Scholar 

  21. J. Shareshian and M. L. Wachs, “Chromatic quasisymmetric functions,” Adv. Math. 295, 497–551 (2016).

    Article  MathSciNet  Google Scholar 

  22. G. M. Ziegler, Lectures on Polytopes (Springer, New York, 1995), Grad. Texts Math. 152.

    Book  Google Scholar 

Download references

Acknowledgments

We are grateful to Kiumars Kaveh for useful conversations. We are also grateful to Shizuo Kaji for communicating to us the results of his computer computations (see Remark 6.7).

Funding

The first author is supported in part by an NSERC Discovery Grant and a Canada Research Chair (Tier 2) Award. She is also grateful to the Osaka City University Advanced Mathematics Institute for support and hospitality during her fruitful visit in Fall 2017, when much of this work was conducted. The second author is partially supported by JSPS Grant-in-Aid for JSPS Fellows 17J04330. The third author is supported in part by JSPS Grant-in-Aid for Scientific Research 16K05152. The fourth author acknowledges the support of the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2018R1A6A3A11047606).

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario, L8S4K1, Canada

    Megumi Harada

  2. Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, 1-5 Yamadaoka, Suita, Osaka, 565-0871, Japan

    Tatsuya Horiguchi

  3. Department of Mathematics, Osaka City University, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan

    Mikiya Masuda

  4. Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology (KAIST), 291 Daehak-ro, Yuseong-gu, Daejeon, 34141, Republic of Korea

    Seonjeong Park

Authors
  1. Megumi Harada
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tatsuya Horiguchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mikiya Masuda
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Seonjeong Park
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Megumi Harada, Tatsuya Horiguchi, Mikiya Masuda or Seonjeong Park.

Additional information

Dedicated to Professor Victor M. Buchstaber on the occasion of his 75th birthday

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2019, Vol. 305, pp. 344–373.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Harada, M., Horiguchi, T., Masuda, M. et al. The Volume Polynomial of Regular Semisimple Hessenberg Varieties and the Gelfand—Zetlin Polytope. Proc. Steklov Inst. Math. 305, 318–344 (2019). https://doi.org/10.1134/S0081543819030192

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543819030192

Search

Navigation