The equivariant cohomology of weighted flag orbifolds

Abstract

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.

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References

  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)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Abe, H., Matsumura, T.: Schur polynomials and weighted Grassmannians. J. Algebr. Comb. 42(3), 875–892 (2015)

    MathSciNet  Article  Google 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)

    MATH  Google Scholar 

  4. 4.

    Atiyah, M.F., Bott, R.: The moment map and equivariant cohomology, Collected Papers 241 (1994)

  5. 5.

    Azam H., Nazir S., Qureshi M. I.: Schubert calculus on weighted symplectic Grassmanians. In Progress

  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)

    MathSciNet  Article  Google 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)

  8. 8.

    Borel, A., Moore, J.C., et al.: Homology theory for locally compact spaces. Mich. Math. J. 7(2), 137–159 (1960)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Bredon, G.E.: Introduction to Compact Transformation Groups, vol. 46. Academic press, Cambridge (1972)

    Google 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)

  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)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Corti, A., Reid, M.: Weighted Grassmannians. In: Algebraic geometry, pp. 141–163. de Gruyter, Berlin (2002)

  14. 14.

    Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties. American Mathematical Soc, Providence (2011)

    Google Scholar 

  15. 15.

    Ehresmann, C.: Sur la topologie de certains espaces homogenes. Ann. Math. 35(2), 396–443 (1934)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Fulton, W.: Young Tableaux: With Applications to Representation Theory and Geometry, vol. 35. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  17. 17.

    Fulton, W., Harris, J.: Representation Theory: A First Course, vol. 129. Springer Science & Business Media, Berlin (2013)

    Google Scholar 

  18. 18.

    Goresky, M., Kottwitz, R., MacPherson, R.: Equivariant cohomology, Koszul duality, and the localization theorem. Invent. Math. 131(1), 25–83 (1997)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Kaji, S.: Schubert calculus, seen from torus equivariant topology. Trends Math. N. Ser. 12(1), 71–90 (2010)

    MathSciNet  Google Scholar 

  21. 21.

    Kaji, S.: Equivariant schubert calculus of Coxeter groups. Proc. Steklov Inst. Math. 275(1), 239–250 (2011)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Kirwan, F.C.: Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, vol. 31. Princeton University Press, Princeton (1984)

    Google 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)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Manivel, L.: Symmetric Functions, Schubert Polynomials, and Degeneracy Loci, vol. 3. American Mathematical Soc., Providence (2001)

    Google Scholar 

  25. 25.

    Qureshi, M. I., Szendrői, B.: Calabi-Yau threefolds in weighted flag varieties. Adv. High Energy Phys. (2012), 14(Art. ID 547317)

  26. 26.

    Qureshi, M.I.: Constructing projective varieties in weighted flag varieties II. Math. Proc. Camb. Phil. Soc. 158, 193–209 (2015)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Qureshi, Muhammad Imran: Computing isolated orbifolds in weighted flag varieties. J. Symb. Comput. 79, Part 2, 457–474 (2017)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Qureshi, M.I., Szendrői, B.: Constructing projective varieties in weighted flag varieties. Bull. Lon. Math Soc. 43(2), 786–798 (2011)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Tymoczko, J.S.: Divided difference operators for partial flag varieties (2009). Preprint arXiv:0912.2545

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Acknowledgements

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.

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Correspondence to Muhammad Imran Qureshi.

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Azam, H., Nazir, S. & Qureshi, M.I. The equivariant cohomology of weighted flag orbifolds. Math. Z. 294, 881–900 (2020). https://doi.org/10.1007/s00209-019-02285-x

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Keywords

  • Weighted flag varieties
  • Equivariant cohomology
  • Schubert classes
  • Double Schubert polynomials