Skip to main content
SpringerLink
Log in

Hessenberg varieties of parabolic type

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

This paper studies the geometry and combinatorics of three interrelated varieties: Springer fibers, Steinberg varieties, and parabolic Hessenberg varieties. We prove that each parabolic Hessenberg variety is the pullback of a Steinberg variety under the projection of the flag variety to an appropriate partial flag variety and we give three applications of this result. The first application constructs an explicit paving of all Steinberg varieties in Lie type A in terms of semistandard tableaux. As a result, we obtain an elementary proof of a theorem of Steinberg and Shimomura that the well-known Kostka numbers count the maximal-dimensional irreducible components of Steinberg varieties. The second application proves an open conjecture for certain parabolic Hessenberg varieties in Lie type A by showing that their Betti numbers equal those of a specific union of Schubert varieties. The third application proves that the irreducible components of parabolic Hessenberg varieties are in bijection with the irreducible components of the Steinberg variety. All three of these applications extend our geometric understanding of the three varieties at the heart of this paper, a full understanding of which is unknown even for Springer varieties, despite over forty years’ worth of work.

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

Access this article

Log in via an institution

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. Springer, T.A.: A construction of representations of Weyl groups. Invent. Math. 44(3), 279–293 (1978)

    Article  MathSciNet  Google Scholar 

  2. Springer, T.A.: Trigonometric sums, Green functions of finite groups and representations of Weyl groups. Invent. Math. 36, 173–207 (1976)

    Article  MathSciNet  Google Scholar 

  3. Fresse, L.: Betti numbers of Springer fibers in type \(A\). J. Algebra 322(7), 2566–2579 (2009)

  4. Tymoczko, J.S.: Linear conditions imposed on flag varieties. Amer. J. Math. 128(6), 1587–1604 (2006)

  5. Spaltenstein, N.: The fixed point set of a unipotent transformation on the flag manifold. Nederl. Akad. Wetensch. Proc. Ser. A =Indag. Math. 38(5):452–456 (1976)

  6. Fresse, L., Melnikov, A.: On the singularity of the irreducible components of a Springer fiber in \(\mathfrak{sl}_n\). Selecta Math. (N.S.), 16(3):393–418 (2010)

  7. Fresse, L.: A unified approach on Springer fibers in the hook, two-row and two-column cases. Transform. Groups 15(2), 285–331 (2010)

  8. Fung, Francis Y. C.: On the topology of components of some Springer fibers and their relation to Kazhdan-Lusztig theory. Adv. Math. 178(2):244–276 (2003)

  9. Wilbert, A.: Topology of two-row Springer fibers for the even orthogonal and symplectic group. Trans. Amer. Math. Soc. 370, 2707–2737 (2018)

  10. Im, M.S., Lai, C-J., Wilbert, A.: Irreducible components of two-row Springer fibers and Nakajima quiver varieties (2019). arXiv:1910.03010

  11. Graham, W., Zierau, R.: Smooth components of Springer fibers. Ann. Inst. Fourier (Grenoble), 61(5):2139–2182 (2012)

  12. De Mari, F., Shayman, M.A.: Generalized Eulerian numbers and the topology of the Hessenberg variety of a matrix. Acta Appl. Math. 12(3), 213–235 (1988)

  13. De Mari, F., Procesi, C., Shayman, M.A.: Hessenberg varieties. Trans. Amer. Math. Soc. 332(2), 529–534 (1992)

    Article  MathSciNet  Google Scholar 

  14. Kostant, B.: Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight \(\rho \). Selecta Math. (N.S.). 2(1):43–91 (1996)

  15. Rietsch, K.: Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties. J. Amer. Math. Soc. 16(2), 363–392 (2003)

  16. Tymoczko, J.S.: Permutation actions on equivariant cohomology of flag varieties. Toric topology, Contemp. Math., Amer. Math. Soc., Providence, RI, vol. 460, pages 365–384 (2008)

  17. Shareshian, J., Wachs, M.L.: Chromatic quasisymmetric functions. Adv. Math. 295, 497–551 (2016)

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

  19. Guay-Paquet, M.: A second proof of the Shareshian-Wachs conjecture, by way of a new Hopf algebra (2016) arXiv:1601.05498

  20. Precup, M.: The Betti numbers of regular Hessenberg varieties are palindromic. Transform. Groups 23(2), 491–499 (2018)

  21. Mbirika, A.: A Hessenberg generalization of the Garsia-Procesi basis for the cohomology ring of Springer varieties. Electron. J. Combin. 17(1):Research Paper 153, 29 (2010)

  22. Abe, H., Harada, M., Horiguchi, T., Masuda, M.: The equivariant cohomology rings of regular nilpotent Hessenberg varieties in lie type A: research announcement. Morfismos 18(2), 51–65 (2014)

  23. Abe, T., Horiguchi, T., Masuda, M., Murai, S., Sato, T.: Hessenberg varieties and hyperplane arrangements. J. Reine Angew. Math. 764, 241–286 (2020)

  24. Steinberg, R.: An occurrence of the Robinson-Schensted correspondence. J. Algebra 113(2), 523–528 (1988)

  25. Borho, W., MacPherson, R.: Partial resolutions of nilpotent varieties. Analysis and topology on singular spaces. II, III (Luminy, 1981), Soc. Math. France, Paris 101, 23–74 (1983)

  26. Fresse, L.: Existence of affine pavings for varieties of partial flags associated to nilpotent elements. Int. Math. Res. Not. IMRN 2, 418–472 (2016)

  27. Shimomura, N.: A theorem on the fixed point set of a unipotent transformation on the flag .manifold. J. Math. Soc. Japan 32(1), 55–64 (1980)

  28. Shimomura, N.: The fixed point subvarieties of unipotent transformations on the flag varieties. J. Math. Soc. Japan 37(3), 537–556 (1985)

  29. Billey, S., Lakshmibai, V.: Singular Loci of Schubert Varieties. Birkhäuser Boston (2000)

  30. Fulton, W.: Young tableaux. London Mathematical Society Student Texts, vol. 35. Cambridge University Press, Cambridge (1997)

  31. Harada, M., Tymoczko, J.: Poset pinball, GKM-compatible subspaces, and Hessenberg varieties. J. Math. Soc. Japan 69(3), 945–994 (2017)

  32. Harada, M., Tymoczko, J.: A positive Monk formula in the \(S^1\)-equivariant cohomology of type \(A\) Peterson varieties. Proc. Lond. Math. Soc. 103(1):40–72 (2011)

  33. Ding, K.: Rook placements and generalized partition varieties. Discrete Math. 176(1–3), 63–95 (1997)

  34. Develin, M., Martin, J.L., Reiner, V.: Classification of Ding’s Schubert varieties: finer rook equivalence. Canad. J. Math. 59(1), 36–62 (2007)

  35. Precup, M., Tymoczko, J.: Springer fibers and schubert points. Eur. J. Combin. 76, 10–26 (2019)

  36. Fulton, W.: Intersection theory. Springer, Berlin (1998)

  37. Precup, Martha: Affine pavings of Hessenberg varieties for semisimple groups. Selecta Math. (N.S.), 19(4):903–922 (2013)

  38. Humphreys, J.E.: Linear algebraic groups. Graduate Texts in Mathematics. Springer, New York-Heidelberg, No. 21 (1975)

  39. Björner, A., Brenti, F.: Combinatorics of Coxeter groups. Graduate Texts in Mathematics, vol. 231. Springer, New York (2005)

  40. Kostant, B.: Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. of Math. 2(74), 329–387 (1961)

  41. Fresse, L.: A notion of inversion number associated to certain quiver flag varieties. Electron. J. Combin. 25(3):Paper 3.41, 29 (2018)

  42. Brundan, J., Ostrik, V.: Cohomology of Spaltenstein varieties. Transform. Groups 16(3), 619–648 (2011)

Download references

Acknowledgements

The first author was partially supported by an AWM-NSF mentoring grant during this work and by National Science Foundation grant DMS-1954001. The second author was partially supported by National Science Foundation grants DMS-1248171 and DMS-1362855.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Washington University in St. Louis, One Brookings Drive, St. Louis, Missouri, 63130, USA

    Martha Precup

  2. Department of Mathematics, Smith College, Northampton, Massachusetts, 01063, USA

    Julianna Tymoczko

Authors
  1. Martha Precup
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Julianna Tymoczko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martha Precup.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Precup, M., Tymoczko, J. Hessenberg varieties of parabolic type. Geom Dedicata 214, 519–542 (2021). https://doi.org/10.1007/s10711-021-00626-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-021-00626-x

Keywords

Mathematics Subject Classification

Search

Navigation