Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

On the quantum product of Schubert classes


Authors: W. Fulton and C. Woodward
Translated by:
Journal: J. Algebraic Geom. 13 (2004), 641-661
DOI: https://doi.org/10.1090/S1056-3911-04-00365-0
Published electronically: February 16, 2004
MathSciNet review: 2072765
Full-text PDF

Abstract | References | Additional Information

Abstract: We give a formula for the smallest powers of the quantum parameters $q$ that occur in a product of Schubert classes in the (small) quantum cohomology of general flag varieties $G/P$. We also include a complete proof of Peterson's quantum version of Chevalley's formula, also for general $G/P$'s.


References [Enhancements On Off] (What's this?)

  • 1. S. Agnihotri and C. Woodward, Eigenvalues of products of unitary matrices and quantum Schubert calculus, Math. Res. Lett., 5(6):817-836, 1998.
  • 2. A. Astashkevich and V. Sadov, Quantum cohomology of partial flag manifolds ${F}\sb{n\sb 1\cdots n\sb k}$, Comm. Math. Phys., 170:503-528,1995.
  • 3. K. Behrend and Yu. Manin, Stacks of stable maps and Gromov-Witten invariants, Duke Math. J., 1:1-60, 1996.
  • 4. P. Belkale. Transformation formulas in Quantum Cohomology, 2001. preprint.
  • 5. A. Bertram, Quantum schubert calculus, Adv. Math., 128:289-305, 1997,
  • 6. A. Bertram, I. Ciocan-Fontanine, and W. Fulton, Quantum multiplication of Schur polynomials, J. Algebra, 219(2):728-746, 1999.
  • 7. A. Bertram, G. Daskalopoulos, and R. Wentworth, Gromov invariants for holomorphic maps from Riemann Surfaces to Grassmannians, J. Amer. Math. Soc., 9:529-571, 1996.
  • 8. A. Borel, Linear algebraic groups, Springer-Verlag, New York, second edition, 1991.
  • 9. R. Bott, A residue formula for holomorphic vector-fields, J. Differential Geometry, 1:311-330, 1967,
  • 10. N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Hermann, Paris, 1968.
  • 11. A. Buch, Quantum cohomology of Grassmannians, Compositio Math. 137 (2003), 227-235.
  • 12. A. Buch, A. Kresch, and H. Tamvakis, Gromov-Witten invariants on Grassmannians, J. Amer. Math. Soc. 16 (2003), 901-915.
  • 13. J. B. Carrell, The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties, In Algebraic groups and their generalizations: classical methods (University Park, PA, 1991), pages 53-61. Amer. Math. Soc., Providence, RI, 1994.
  • 14. C. Chevalley, Sur les décompositions cellulaires des espaces ${G}/{B}$, In Algebraic groups and their generalizations: classical methods (University Park, PA, 1991), pages 1-23. Amer. Math. Soc., Providence, RI, 1994, With a foreword by Armand Borel.
  • 15. L. Chen, Quantum cohomology of flag manifolds, Adv. Math. 174 (2003), 1-34.
  • 16. I. Ciocan-Fontanine, The quantum cohomology ring of flag varieties, Trans. Amer. Math. Soc., 351(7):2695-2729, 1999.
  • 17. I. Ciocan-Fontanine, On quantum cohomology rings of partial flag varieties, Duke Math. J., 98(3):485-524, 1999.
  • 18. S. Fomin, S. Gelfand, and A. Postnikov, Quantum Schubert polynomials, J. Amer. Math. Soc., 10:565-596, 1997.
  • 19. W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc., 37:209-249, 2000.
  • 20. W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, In Algebraic geometry--Santa Cruz 1995, pages 45-96. Amer. Math. Soc., Providence, RI, 1997.
  • 21. A. Givental and Bumsig Kim, Quantum cohomology of flag manifolds and Toda lattices, Comm. Math. Phys., 168(3):609-641, 1995.
  • 22. A. Grothendieck. Sur la classification des fibrés holomorphes sur la sphère de Riemann. Amer. J. Math., 79:121-138, 1957.
  • 23. A. Hirschowitz, Le groupe de Chow équivariant, C. R. Acad. Sci. Paris Sér. I Math., 298(5):87-89, 1984.
  • 24. J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, Cambridge, 1990.
  • 25. B. Kim and R. Pandharipande. The connectedness of the moduli space of maps to homogeneous spaces. In Symplectic geometry and mirror symmetry (Seoul, 2000), pages 187-201. World Sci. Publishing, River Edge, NJ, 2001.
  • 26. Bumsig Kim, Quantum cohomology of flag manifolds ${G}/{B}$ and quantum Toda lattices, Ann. of Math. (2), 149(1):129-148, 1999.
  • 27. Bumsig Kim, Quantum cohomology of partial flag manifolds and a residue formula for their intersection pairings, Internat. Math. Res. Notices, 1:1-15 (electronic), 1995.
  • 28. S. L. Kleiman, The transversality of a general translate, Compositio Math., 28:287-297, 1974.
  • 29. J. Kollár, Rational curves on algebraic varieties, Springer-Verlag, Berlin, 1996.
  • 30. M. Kontsevich, Enumeration of rational curves via torus actions, In The moduli space of curves (Texel Island, 1994), pages 335-368. Birkhäuser Boston, Boston, MA, 1995.
  • 31. M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys., 164(3):525-562, 1994.
  • 32. A. Kresch and H. Tamvakis, Quantum cohomology of the Lagrangian Grassmannian, J. Algebraic Geom. 12 (2003), 777-810.
  • 33. D. Peterson, Lectures on quantum cohomology of G/P, M.I.T., 1996.
  • 34. Alexander Postnikov. Affine approach to quantum Schubert calculus. math.CO/ 0205165.
  • 35. B. Siebert and G. Tian, On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator, Asian J. Math., 1:679-695, 1997.
  • 36. J. F. Thomsen, Irreducibility of $\overline{{M}}\sb {0,n}({G}/{P},\beta)$, Internat. J. Math., 9(3):367-376, 1998.
  • 37. E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, In Geometry, topology, and physics, volume VI of Conf. Proc. Lecture Notes Geom. Topology, Berkeley, 1989, 1995. Internat. Press, Cambridge, 1995.
  • 38. C. Woodward. On D. Peterson's comparison formula for Gromov-Witten invariants of ${G}/{P}$. math.AG/0206073.
  • 39. A. Yong, On bounds for quantum multiplication, Proc. Amer. Math. Soc. 131 (2003), 2649-2655.


Additional Information

W. Fulton
Affiliation: Department of Mathematics, University of Michigan, 2074 East Hall, Ann Arbor, Michigan 48109-1109
Email: wfulton@math.lsa.umich.edu

C. Woodward
Affiliation: Mathematics-Hill Center, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019
Email: ctw@math.rutgers.edu

DOI: https://doi.org/10.1090/S1056-3911-04-00365-0
Received by editor(s): April 8, 2002
Published electronically: February 16, 2004
Additional Notes: The first author was partially supported by NSF grant DMS9970435. The second author was partially supported by NSF grant DMS9971357.

American Mathematical Society