Skip to main content
SpringerLink
Log in

Quantum hooks and mirror symmetry for flag varieties

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

Given a flag variety \({{\,\textrm{Fl}\,}}(n;r_1, \dots , r_\rho )\), there is natural ring morphism from the symmetric polynomial ring in \(r_1\) variables to the quantum cohomology of the flag variety. In this paper, we show that for a large class of partitions \(\lambda \), the image of \(s_\lambda \) under the ring homomorphism is a Schubert class which is described by partitioning \(\lambda \) into a quantum hook (or q-hook) and a tuple of smaller partitions. We use this result to show that the Plücker coordinate mirror of the flag variety describes quantum cohomology relations. This gives new insight into the structure of this superpotential, and the relation between superpotentials of flag varieties and those of Grassmannians (where the superpotential was introduced by Marsh–Rietsch).

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Batyrev, V.V., Ciocan-Fontanine, I., Kim, B., van Straten, D.: Mirror symmetry and toric degenerations of partial flag manifolds. Acta Math. 184(1), 1–39 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bertram, A., Ciocan-Fontanine, I., Fulton, W.: Quantum multiplication of Schur polynomials. J. Algebra 219(2), 728–746 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Billey, S.C., Jockusch, W., Stanley, R.P.: Some combinatorial properties of Schubert polynomials. J. Algebr. Combin. 2(4), 345–374 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, L., Gibney, A., Heller, L., Kalashnikov, E., Larson, H., Xu, W.: On an equivalence of divisors on \(\overline{\text{ M }}_{0, n}\) from Gromov–Witten theory and conformal blocks. Transform. Groups 20, 20 (2022)

    Google Scholar 

  5. Ciocan-Fontanine, I.: On quantum cohomology rings of partial flag varieties. Duke Math. J. 98(3), 485–524 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Coates, T., Corti, A., Galkin, S., Golyshev, V., Kasprzyk, A.: Mirror Symmetry and Fano Manifolds. European Congress of Mathematics, pp. 285–300. European Mathematical Society, Zürich (2013)

    MATH  Google Scholar 

  7. Eguchi, T., Hori, K., Xiong, C.: Gravitational quantum cohomology. Int. J. Mod. Phys. A 12, 1743–1782 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fomin, S., Gelfand, S., Postnikov, A.: Quantum Schubert polynomials. J. Am. Math. Soc 10, 565–596 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fulton, W., Pandharipande, R.: Notes on stable maps and quantum cohomology, Algebraic geometry—Santa Cruz (1995). In: Proceedings of Symposia in Pure Mathematics, vol. 62. American Mathematical Society, Providence, pp. 45–96 (1997)

  10. Givental, A.: A mirror theorem for toric complete intersections, topological field theory, primitive forms and related topics (Kyoto, 1996). Progress in Mathematics, vol. 160. Birkhäuser, Boston, pp. 141–175 (1998)

  11. Gu, W., Kalashnikov, E.: A rim-hook rule for quiver flag varieties, p. 9 (2020). arXiv:2009.02810

  12. Hori, K., Vafa, C.: Mirror symmetry (2000). arXiv:hep-th/0002222

  13. Kalashnikov, E.: A Plücker coordinate mirror for type A flag varieties. Bull. Lond. Math. Soc. 54(4), 1308–1325 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kim, B.: On equivariant quantum cohomology. Int. Math. Res. Not. 1996(17), 841–851 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Knutson, A., Miller, E., Yong, A.: Gröbner geometry of vertex decompositions and of flagged tableaux. J. Reine Angew. Math. 630, 1–31 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kontsevich, M.: Enumeration of rational curves via torus actions, the moduli space of curves (Texel Island, 1994). Progress in Mathematics, vol. 129. Birkhäuser, Boston, pp. 335–368 (1995)

  17. Lascoux, A., Schützenberger, M.-P.: Polynômes de Schubert. C. R. Acad. Sci. Paris Sér. I Math. 294(13), 447–450 (1982)

    MathSciNet  MATH  Google Scholar 

  18. Lian, B., Liu, K., Yau, S.T.: Mirror principle I. Asian J. Math. 1(4), 729–763 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Marsh, R.J., Rietsch, K.: The B-model connection and mirror symmetry for Grassmannians. Adv. Math. 366, 107027 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rietsch, K., Williams, L.: Newton–Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians. Duke Math. J. 168(18), 3437–3527 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Rietsch, K.: A mirror symmetric construction of qHT(G/P)(q). Adv. Math. 217(6), 2401–2442 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Scott, J.S.: Grassmannians and cluster algebras. Proc. Lond. Math. Soc. 92(2), 345–380 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Spacek, P.: Laurent polynomial Landau–Ginzburg models for cominiscule homogeneous spaces. Transform. Groups 20, 20 (2021)

    Google Scholar 

  24. Spacek, P., Wang, C.: Towards Landau-Ginzburg models for cominuscule spaces via the exceptional cominuscule family. arXiv:2204.03548 (2022)

Download references

Acknowledgements

The authors would like to thank Konstanze Rietsch, Dave Anderson, and Jennifer Morse for helpful conversations.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Swarthmore College, Swarthmore, PA, 19081, USA

    L. Chen

  2. Department of Pure Mathematics, University of Waterloo, Waterloo, N2L 3G1, Canada

    E. Kalashnikov

Authors
  1. L. Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Kalashnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Kalashnikov.

Additional information

Publisher's Note

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

L. Chen was partially supported by Simons Collaboration Grant 524354 and NSF Grant DMS-2101861. E. Kalashnikov is supported by an NSERC Discovery Grant.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, L., Kalashnikov, E. Quantum hooks and mirror symmetry for flag varieties. Math. Z. 305, 28 (2023). https://doi.org/10.1007/s00209-023-03359-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00209-023-03359-7

Search

Navigation