The equivariant cohomology of weighted flag orbifolds

  • Haniya Azam
  • Shaheen Nazir
  • Muhammad Imran QureshiEmail author


We describe the torus-equivariant cohomology of weighted partial flag orbifolds \({\mathrm {w}}\Sigma \) of type A. We establish counterparts of several results known for the partial flag variety that collectively constitute what we refer to as “Schubert Calculus on \({\mathrm {w}}\Sigma \) ”. For the weighed Schubert classes in \({\mathrm {w}}\Sigma \), we give the Chevalley’s formula. In addition, we define the weighted analogue of double Schubert polynomials and give the corresponding Chevalley–Monk’s formula.


Weighted flag varieties Equivariant cohomology Schubert classes Double Schubert polynomials 



We wish to thank Waqar Ali Shah for several helpful discussions. We also thank Frank Sottile and Balázs Szendrői for their comments on earlier drafts of this article. Thanks are also due to Shizu Kaji, Allen Knutson and Loring Tu for some useful conversations. Last but not least, we are indebted to the anonymous referee for his/her comments and suggestions which significantly improved the earlier version of this paper. HA and MIQ were supported by the HEC’s NRPU research grant “5906/Punjab/NRPU/R&D/HEC/2016”. MIQ was on a fellowship of Alexander–Von–Humboldt foundation during a part of this paper.


  1. 1.
    Abe, H., Matsumura, T.: Equivariant cohomology of weighted Grassmannians and weighted Schubert classes. Int. Math. Res. Not. IMRN 2015(9), 2499–2524 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abe, H., Matsumura, T.: Schur polynomials and weighted Grassmannians. J. Algebr. Comb. 42(3), 875–892 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alberto, A.: Cohomologie t-equivariante de \(G/B\) pour an groupe \(G\) de kac-moody. C.R. Acad. Sci. Paris 302, 631–634 (1986)zbMATHGoogle Scholar
  4. 4.
    Atiyah, M.F., Bott, R.: The moment map and equivariant cohomology, Collected Papers 241 (1994)Google Scholar
  5. 5.
    Azam H., Nazir S., Qureshi M. I.: Schubert calculus on weighted symplectic Grassmanians. In ProgressGoogle Scholar
  6. 6.
    Berline, N., Vergne, M., et al.: Zéros d’un champ de vecteurs et classes caractéristiques équivariantes. Duke Math. J. 50(2), 539–549 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Borel, A., Bredon, G., Floyd, E.E., Montgomery, D., Palais, R.: Seminar on Transformation Groups, vol. AM-46. Princeton University Press, Princeton (1960)Google Scholar
  8. 8.
    Borel, A., Moore, J.C., et al.: Homology theory for locally compact spaces. Mich. Math. J. 7(2), 137–159 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bredon, G.E.: Introduction to Compact Transformation Groups, vol. 46. Academic press, Cambridge (1972)zbMATHGoogle Scholar
  10. 10.
    Brion, M.: Lectures on the Geometry of Flag Varieties. In: Topics in Cohomological Studies of Algebraic Varieties, pp. 33–85. Springer, Berlin (2005)Google Scholar
  11. 11.
    Brown, G., Alexander, K., Lei, Z.: Gorenstein formats, canonical and Calabi-Yau threefolds (2014). Preprint arXiv:1409.4644
  12. 12.
    Chang, T., Skjelbred, T.: Topological schur lemma and related results. Bull. Am. Math. Soc. 79(5), 1036–1038 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Corti, A., Reid, M.: Weighted Grassmannians. In: Algebraic geometry, pp. 141–163. de Gruyter, Berlin (2002)Google Scholar
  14. 14.
    Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties. American Mathematical Soc, Providence (2011)CrossRefzbMATHGoogle Scholar
  15. 15.
    Ehresmann, C.: Sur la topologie de certains espaces homogenes. Ann. Math. 35(2), 396–443 (1934)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fulton, W.: Young Tableaux: With Applications to Representation Theory and Geometry, vol. 35. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  17. 17.
    Fulton, W., Harris, J.: Representation Theory: A First Course, vol. 129. Springer Science & Business Media, Berlin (2013)zbMATHGoogle Scholar
  18. 18.
    Goresky, M., Kottwitz, R., MacPherson, R.: Equivariant cohomology, Koszul duality, and the localization theorem. Invent. Math. 131(1), 25–83 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Guillemin, V., Holm, T., Zara, C.: A GKM description of the equivariant cohomology ring of a homogeneous space. J. Algebr. Comb. 23(1), 21–41 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kaji, S.: Schubert calculus, seen from torus equivariant topology. Trends Math. N. Ser. 12(1), 71–90 (2010)MathSciNetGoogle Scholar
  21. 21.
    Kaji, S.: Equivariant schubert calculus of Coxeter groups. Proc. Steklov Inst. Math. 275(1), 239–250 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kirwan, F.C.: Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, vol. 31. Princeton University Press, Princeton (1984)zbMATHGoogle Scholar
  23. 23.
    Kostant, B., Kumar, S.: The nil Hecke ring and cohomology of gp for a kac-moody group g. Adv. Math. 62(3), 187–237 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Manivel, L.: Symmetric Functions, Schubert Polynomials, and Degeneracy Loci, vol. 3. American Mathematical Soc., Providence (2001)zbMATHGoogle Scholar
  25. 25.
    Qureshi, M. I., Szendrői, B.: Calabi-Yau threefolds in weighted flag varieties. Adv. High Energy Phys. (2012), 14(Art. ID 547317)Google Scholar
  26. 26.
    Qureshi, M.I.: Constructing projective varieties in weighted flag varieties II. Math. Proc. Camb. Phil. Soc. 158, 193–209 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Qureshi, Muhammad Imran: Computing isolated orbifolds in weighted flag varieties. J. Symb. Comput. 79, Part 2, 457–474 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Qureshi, M.I.: Polarized 3-folds in a codimension 10 weighted homogeneous F-4 variety. J. Geom. Phys. 120, 52–61 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Qureshi, M.I., Szendrői, B.: Constructing projective varieties in weighted flag varieties. Bull. Lon. Math Soc. 43(2), 786–798 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Tymoczko, J.S.: Divided difference operators for partial flag varieties (2009). Preprint arXiv:0912.2545

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Haniya Azam
    • 1
  • Shaheen Nazir
    • 1
  • Muhammad Imran Qureshi
    • 1
    • 2
    Email author
  1. 1.Department of Mathematics, SBASSELahore University of Management Sciences (LUMS)LahorePakistan
  2. 2.Mathematisches InstitutUniversität TübingenTübingenGermany

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