Advertisement

Mathematische Zeitschrift

, Volume 291, Issue 3–4, pp 1569–1603 | Cite as

Equivariant quantum cohomology of the odd symplectic Grassmannian

  • Leonardo C. Mihalcea
  • Ryan M. ShiflerEmail author
Article
  • 31 Downloads

Abstract

The odd symplectic Grassmannian \(\mathrm {IG}:=\mathrm {IG}(k, 2n+1)\) parametrizes k dimensional subspaces of \({\mathbb {C}}^{2n+1}\) which are isotropic with respect to a general (necessarily degenerate) symplectic form. The odd symplectic group acts on \(\mathrm {IG}\) with two orbits, and \(\mathrm {IG}\) is itself a smooth Schubert variety in the submaximal isotropic Grassmannian \(\mathrm {IG}(k, 2n+2)\). We use the technique of curve neighborhoods to prove a Chevalley formula in the equivariant quantum cohomology of \(\mathrm {IG}\), i.e. a formula to multiply a Schubert class by the Schubert divisor class. This generalizes a formula of Pech in the case \(k=2\), and it gives an algorithm to calculate any multiplication in the equivariant quantum cohomology ring.

Mathematics Subject Classification

Primary 14N35 Secondary 14N15 14M15 

Notes

Acknowledgements

We would like to thank Dan Orr and Mark Shimozono for discussions and valuable suggestions and to Pierre-Emmanuel Chaput, Changzheng Li, and Nicolas Perrin for discussions and collaborations on related projects. Special thanks are due to Anders Buch for encouragement and interest in this project. We thank the referee for carefully reading our paper.

References

  1. 1.
    Anderson, D.: Introduction to equivariant cohomology in algebraic geometry. Contributions to algebraic geometry, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, pp. 71–92 (2012)Google Scholar
  2. 2.
    Björner, A., Brenti, F.: Combinatorics of Coxeter groups. Graduate texts in mathematics, vol 231. Springer, New York, p. xiv+363 (2005)Google Scholar
  3. 3.
    Brion, M.: Lectures on the geometry of flag varieties. In: Topics in cohomological studies of algebraic varieties. Birkhäuser, Basel, Trends Math, pp. 33–85 (2005)Google Scholar
  4. 4.
    Buch, A., Chaput, P.E., Mihalcea, L.C., Perrin, N.: A Chevalley formula for the equivariant quantum \(K\)-theory of cominuscule varieties, to appear in Algebraic Geometry. http://arxiv.org/pdf/1604.07500.pdf
  5. 5.
    Buch, A., Chaput, P.E., Mihalcea, L.C., Perrin, N.: Finiteness of cominuscule quantum K-theory. Annales Sci. de L’École Normale Supérieure, no. 46 (2013)Google Scholar
  6. 6.
    Buch, A.S.: Quantum cohomology of Grassmannians. Compos. Math. 137(2), 227–235 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kresch, A., Tamvakis, H.: Quantum Pieri rules for isotropic Grassmannians. Invent. Math. 178(2), 345–405 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Buch, A., Kresch, A., Tamvakis, H.: Quantum Giambelli formulas for isotropic Grassmannians. Math. Ann. 354(3), 801–812 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Buch, A.S., Mihalcea, L.C.: Curve neighborhoods of Schubert varieties. J. Diff. Geom. 99(2), 255–283 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cheong, D., Li, C.: On the conjecture O of GGI for G/P. Adv. Math. 306, 704–721 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chevalley, C.: Sur les décompositions cellulaires des espaces \(G/B\), Algebraic groups and their generalizations: classical methods (University Park, PA, 1991), Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc., Providence, RI, 1994, With a foreword by Armand Borel, pp. 1–23 (1994)Google Scholar
  12. 12.
    Fulton, W., Pandharipande, R.: Notes on stable maps and quantum cohomology, Algebraic geometry–Santa Cruz: Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc. Providence, RI 1997, pp. 45–96 (1995)Google Scholar
  13. 13.
    Galkin, S., Golyshev, V., Iritani, H.: Gamma classes and quantum cohomology of Fano manifolds: gamma conjectures. Duke Math. J. 165(11), 2005–2077 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gel’fand, I.M., Zelevinskiĭ, A.V.: Models of representations of classical groups and their hidden symmetries. Funktsional. Anal. i Prilozhen. 18(3), 14–31 (1984)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Graber, T.: Enumerative geometry of hyperelliptic plane curves. J. Algebraic Geom. 10(4), 725–755 (1998)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Graham, W.: Positivity in equivariant Schubert calculus. Duke Math. J. 109(3), 599–614 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hartshorne, R.: Algebraic geometry. Graduate Texts in Mathematics., 52nd edn. Springer, New York-Heidelberg (1977)CrossRefGoogle Scholar
  18. 18.
    Humphreys, J.E.: Linear algebraic groups. Springer, New York-Heidelberg. Graduate Texts in Mathematics, No. 21 (1975)Google Scholar
  19. 19.
    Ikeda, T., Mihalcea, L.C., Naruse, H.: Factorial \(P\)- and \(Q\)-Schur functions represent equivariant quantum Schubert classes. Osaka J. Math. 53(3), 591–619 (2016)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Kim, B., Pandharipande, R.: The connectedness of the moduli space of maps to homogeneous spaces. In: Symplectic geometry and mirror symmetry (Seoul: 2000), World Sci. Publ. River Edge, NJ 2001, pp. 187–201 (2000)Google Scholar
  21. 21.
    Knutson, A., Tao, T.: Puzzles and (equivariant) cohomology of Grassmannians. Duke Math. J. 119(2), 221–260 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kostant, B., Kumar, S.: \(T\)-equivariant \(K\)-theory of generalized flag varieties. Proc. Nat. Acad. Sci. USA 84(13), 4351–4354 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kresch, A., Tamvakis, H.: Quantum cohomology of the Lagrangian Grassmannian. J. Algebraic Geom. 12(4), 777–810 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kumar, S.: Kac-Moody groups, their flag varieties and representation theory. Progress in Mathematics, vol. 204. Birkhäuser Boston Inc, Boston, MA (2002)Google Scholar
  25. 25.
    Li, C., Mihalcea, L.C., Shifler, R.: Conjecture O holds for the odd-symplectic Grassmannian. arXiv:1706.00744
  26. 26.
    Mare, A., Mihalcea, L.: An affine deformation of the quantum cohomology ring of flag manifolds and periodic Toda lattice, to appear in Proc. of London Math. Soc. http://arxiv.org/pdf/1409.3587.pdf
  27. 27.
    Matsumura, H.: Commutative ring theory: Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1986). (Translated from the Japanese by M. Reid) Google Scholar
  28. 28.
    Mihai, I.A.: Odd symplectic flag manifolds. Transform. Groups 12(3), 573–599 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Mihai, I.: Variétés de drapeaux symplectiques impaires. Ph.D. thesis, Institut Fourier (2005)Google Scholar
  30. 30.
    Mihalcea, L.C.: Equivariant quantum Schubert calculus. Adv. Math. 203(1), 1–33 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Mihalcea, L.C.: Positivity in equivariant quantum Schubert calculus. Am. J. Math. 128(3), 787–803 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Mihalcea, L.C.: On equivariant quantum cohomology of homogeneous spaces: Chevalley formulae and algorithms. Duke Math. J. 140(2), 321–350 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Mihalcea, L.C.: Giambelli formulae for the equivariant quantum cohomology of the Grassmannian. Trans. Am. Math. Soc. 360(5), 2285–2301 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Molev, A.I., Sagan, B.E.: A Littlewood-Richardson rule for factorial Schur functions. Trans. Am. Math. Soc. 351(11), 4429–4443 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Okounkov, A., Olshanskiĭ, G.: Shifted Schur functions. Algebra i Analiz 9(2), 73–146 (1997)MathSciNetGoogle Scholar
  36. 36.
    Okounkov, A.: Quantum immanants and higher Capelli identities. Transform. Groups 1(1–2), 99–126 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Pech, C.: Cohomologie quantique des grassmanniennes symplectiques impaire. Ph.D. thesis, Université de Grenoble (2011)Google Scholar
  38. 38.
    Pech, C.: Quantum cohomology of the odd symplectic Grassmannian of lines. J. Algebra 375, 188–215 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Robert, A.: Proctor, odd symplectic groups. Invent. Math. 92(2), 307–332 (1988)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Tamvakis, H.: Schubert polynomials and degeneracy locus formulas. https://arxiv.org/pdf/1602.05919.pdf
  41. 41.
    Tamvakis, H., Wilson, E.: Double theta polynomials and equivariant Giambelli formulas. Math. Proc. Cambridge Philos. Soc. 160(2), 353–377 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Thomsen, J.F.: Irreducibility of \(\overline{M}_{0, n}(G/P,\beta )\). Int. J. Math. 9(3), 367–376 (1998)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, 460 McBryde HallVirginia TechBlacksburgUSA
  2. 2.Department of Mathematics and Computer Science, Henson Science HallSalisbury UniversitySalisburyUSA

Personalised recommendations