Skip to main content

Structure constants for Chern classes of Schubert cells


A formula for the structure constants of the multiplication of Schubert classes is obtained in (Rebecca and Allen. arXiv preprint arXiv:1909.05283, 2019). In this note, we prove analogous formulae for the Chern–Schwartz–MacPherson (CSM) classes and Segre–Schwartz–MacPherson (SSM) classes of Schubert cells in the flag variety. By the equivalence between the CSM classes and the stable basis elements for the cotangent bundle of the flag variety, a formula for the structure constants for the latter is also deduced.

This is a preview of subscription content, access via your institution.


  1. If X is not smooth, we can embed X into a smooth ambient space, and use the total Chern class of the ambient space to define the SSM classes, see [2].

  2. These operators are the \(\hbar =1\) specialization of \(L_i\) and \(L^\vee _i\) in [3, Section 5.2].

  3. Here we have used the fact \((-1)^{\dim G/B}e^{T\times {\mathbb C}^*}(T^*(G/B))|_{\hbar =1}=c^T(T(G/B))\), see the proof of Corollary 7.4 in loc. cit..

  4. The author thanks L. Mihalcea for pointing out this proof.


  1. Aluffi, P., Mihalcea, L.C., Schürmann, J., Su, C.: Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman’s problem. arXiv preprint arXiv:1902.10101 (2019)

  2. Aluffi, P., Mihalcea, L.C., Schürmann, J., Su, C.: Positivity of Segre-Macpherson classes. arXiv preprint arXiv:1902.00762 (2019)

  3. Aluffi, P., Mihalcea, L.C., Schürmann, J., Su, C.: Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-Macpherson classes of Schubert cells. arXiv preprint arXiv:1709.08697 (2017)

  4. Aluffi, P., Mihalcea, L.: Chern classes of Schubert cells and varieties. J. Algebr. Geom. 18(1), 63–100 (2009)

    MathSciNet  Article  Google Scholar 

  5. Aluffi, P., Mihalcea, L.C.: Chern-Schwartz-Macpherson classes for Schubert cells in flag manifolds. Compos. Math. 152(12), 2603–2625 (2016)

    MathSciNet  Article  Google Scholar 

  6. Andersen, H.H., Jantzen, J.C., Soergel, W.: Representations of quantum groups at a p-th root of unity and of semisimple groups in characteristic p: independence of p. Société mathématique de France (1994)

  7. Anderson, D., Griffeth, S., Miller, E.: Positivity and Kleiman transversality in equivariant K-theory of homogeneous spaces. arXiv preprint arXiv:0808.2785 (2008)

  8. Bernsteĭn, I.N., Gelfand, I.M., Gelfand, S.I.: Schubert cells, and the cohomology of the spaces \(G/P\). Uspehi Mat. Nauk 28(3(171)), 3–26 (1973)

    MathSciNet  Google Scholar 

  9. Billey, S.C.: Kostant polynomials and the cohomology ring for G/B. Duke Math. J. 96(1), 205 (1999)

    MathSciNet  Article  Google Scholar 

  10. Brasselet, J.-P., Schürmann, J., Yokura, S.: Hirzebruch classes and motivic Chern classes for singular spaces. J. Topol. Anal. 2(01), 1–55 (2010)

    MathSciNet  Article  Google Scholar 

  11. Brion, M.: Positivity in the Grothendieck group of complex flag varieties. J. Algebra 258(1), 137–159 (2002)

    MathSciNet  Article  Google Scholar 

  12. Buch, A.S.: A Littlewood-Richardson rule for the K-theory of Grassmannians. Acta Math. 189(1), 37–78 (2002)

    MathSciNet  Article  Google Scholar 

  13. Buch, A.S., Kresch, A., Purbhoo, K., Tamvakis, H.: The puzzle conjecture for the cohomology of two-step flag manifolds. J. Algebr. Comb. 44(4), 973–1007 (2016)

    MathSciNet  Article  Google Scholar 

  14. Changjian, S.: Restriction formula for stable basis of the Springer resolution. Sel. Math. 23(1), 497–518 (2017)

    MathSciNet  Article  Google Scholar 

  15. Chriss, N., Ginzburg, V.: Representation theory and complex geometry. Springer Science & Business Media, New York (2009)

    MATH  Google Scholar 

  16. Collins, V.: A puzzle formula for \(H^*_{T\times \mathbb{C}^*}(T^*\mathbb{P}^n)\). Séminaire Lotharingien de Combinatoire 78B, (2017)

  17. Feher, L.M., Rimanyi, R., Weber, A.: Motivic Chern classes and K-theoretic stable envelopes. arXiv preprint arXiv:1802.01503 (2018)

  18. Goldin, R., Knutson, A.: Schubert structure operators and \({K_T(G/B)}\). arXiv preprint arXiv:1909.05283 (2019)

  19. Graham, W.: Positivity in equivariant Schubert calculus. Duke Math. J. 109(3), 599–614 (2001)

    MathSciNet  Article  Google Scholar 

  20. Huh, J.: Positivity of Chern classes of Schubert cells and varieties. J. Algebr. Geom. 25(1), 177–199 (2016)

    MathSciNet  Article  Google Scholar 

  21. Knutson, Allen, Zinn-Justin, Paul: in preparation

  22. Knutson, A., Zinn-Justin, P.: Schubert puzzles and integrability I: invariant trilinear forms. arXiv preprint arXiv:1706.10019 (2017)

  23. Knutson, A., Tao, T.: Puzzles and (equivariant) cohomology of Grassmannians. Duke Math. J. 119(2), 221–260 (2003)

    MathSciNet  Article  Google Scholar 

  24. Kumar, S.: Positivity in T-equivariant T-theory of flag varieties associated to Kac-Moody groups. J. Eur. Math. Soc. 19(8), 2469–2519 (2017)

    MathSciNet  Article  Google Scholar 

  25. Lusztig, G.: Affine Hecke algebras and their graded version. J. Am. Math. Soc. 2(3), 599–635 (1989)

    MathSciNet  Article  Google Scholar 

  26. MacPherson, R.D.: Chern classes for singular algebraic varieties. Ann. Math., pages 423–432 (1974)

  27. Maulik, D., Okounkov, A.: Quantum groups and quantum cohomology. Astérisque, (408) (2019)

  28. Mihalcea, L., Withrow, C.: in preparation

  29. Ohmoto, T.: Equivariant Chern classes of singular algebraic varieties with group actions. In: Mathematical proceedings of the cambridge philosophical society, volume 140, pages 115–134. Cambridge University Press (2006)

  30. Okounkov, A.: Enumerative geometry and geometric representation theory. Algebraic Geometry: Salt Lake City, pages 419–457 (2015)

  31. Okounkov, A.: On the crossroads of enumerative geometry and geometric representation theory. arXiv preprint arXiv:1801.09818, (2018)

  32. Okounkov, A.: Lectures on K-theoretic computations in enumerative geometry. Geometry of moduli spaces and representation theory, IAS/Park City. Math. Ser. 24, 251–380 (2017)

    MATH  Google Scholar 

  33. Rimányi, R., Varchenko, A.: Equivariant Chern-Schwartz-Macpherson classes in partial flag varieties: interpolation and formulae. arXiv preprint arXiv:1509.09315 (2015)

  34. Rimányi, R., Tarasov, V., Varchenko, A.: Partial flag varieties, stable envelopes, and weight functions. Quantum Topol. 6(2), 333–364 (2015)

    MathSciNet  Article  Google Scholar 

  35. Schürmann, J.: Chern classes and transversality for singular spaces. In: Singularities in Geometry, Topology, Foliations and Dynamics, pages 207–231. Springer (2017)

  36. Schwartz, M.-H.: Classes caractéristiques définies par une stratification d’une variété analytique complexe. I. C. R. Acad. Sci. Paris 260, 3262–3264 (1965)

    MathSciNet  MATH  Google Scholar 

  37. Schwartz, M.-H.: Classes caractéristiques définies par une stratification d’une variété analytique complexe. II. C. R. Acad. Sci. Paris 260, 3535–3537 (1965)

    MathSciNet  MATH  Google Scholar 

  38. Su, C., Zhao, G., Zhong, C.: On the K-theory stable bases of the Springer resolution. arXiv preprint arXiv:1708.08013 (2017)

  39. Su, C., Zhong, C.: Stable bases of the Springer resolution and representation theory. arXiv preprint arXiv:1904.06613 (2019)

  40. Su, C.: Motivic Chern classes and Iwahori invariants of principal series. To appear in Proceedings of International Congress of Chinese Mathematicians (2019)

  41. Varagnolo, M.: Quiver varieties and Yangians. Lett. Math. Phys. 53(4), 273–283 (2000)

    MathSciNet  Article  Google Scholar 

  42. Willems, M.: A Chevalley formula in equivariant K-theory. J. Algebra 308(2), 764–779 (2007)

    MathSciNet  Article  Google Scholar 

Download references


The author thanks A. Knutson and A. Yong for discussions. Special thanks go to L. Mihalcea for providing the proof of Theorem 2.5. The author also thanks the anonymous referees for useful comments.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Changjian Su.

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

Verify currency and authenticity via CrossMark

Cite this article

Su, C. Structure constants for Chern classes of Schubert cells. Math. Z. 298, 193–213 (2021).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • MacPherson classes
  • Flag variety
  • Bott–Samelson variety

Mathematics Subject Classification

  • 57T15
  • 05E15