On the quantum product of Schubert classes
Authors:
W. Fulton and C. Woodward
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
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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.
ag:ei S. Agnihotri and C. Woodward, Eigenvalues of products of unitary matrices and quantum Schubert calculus, Math. Res. Lett., 5(6):817–836, 1998.
as:qc A. Astashkevich and V. Sadov, Quantum cohomology of partial flag manifolds ${F}_{n_ 1\cdots n_ k}$, Comm. Math. Phys., 170:503–528,1995.
bm:st K. Behrend and Yu. Manin, Stacks of stable maps and Gromov-Witten invariants, Duke Math. J., 1:1–60, 1996.
be:tr P. Belkale. Transformation formulas in Quantum Cohomology, 2001. preprint.
be:qs A. Bertram, Quantum schubert calculus, Adv. Math., 128:289–305, 1997,
be:qm A. Bertram, I. Ciocan-Fontanine, and W. Fulton, Quantum multiplication of Schur polynomials, J. Algebra, 219(2):728–746, 1999.
be:gi 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.
bo:lag A. Borel, Linear algebraic groups, Springer-Verlag, New York, second edition, 1991.
bo:rf R. Bott, A residue formula for holomorphic vector-fields, J. Differential Geometry, 1:311–330, 1967,
bo:lg 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.
bu:qc A. Buch, Quantum cohomology of Grassmannians, Compositio Math. 137 (2003), 227–235.
bkt:lg A. Buch, A. Kresch, and H. Tamvakis, Gromov-Witten invariants on Grassmannians, J. Amer. Math. Soc. 16 (2003), 901–915.
ca:de 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.
ch:su 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.
ch:qf L. Chen, Quantum cohomology of flag manifolds, Adv. Math. 174 (2003), 1–34.
cf:qc2 I. Ciocan-Fontanine, The quantum cohomology ring of flag varieties, Trans. Amer. Math. Soc., 351(7):2695–2729, 1999.
cf:qc3 I. Ciocan-Fontanine, On quantum cohomology rings of partial flag varieties, Duke Math. J., 98(3):485–524, 1999.
fgp:qs S. Fomin, S. Gelfand, and A. Postnikov, Quantum Schubert polynomials, J. Amer. Math. Soc., 10:565–596, 1997.
fu:ev W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc., 37:209–249, 2000.
fp:st 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.
gi:qc A. Givental and Bumsig Kim, Quantum cohomology of flag manifolds and Toda lattices, Comm. Math. Phys., 168(3):609–641, 1995.
gr:cl A. Grothendieck. Sur la classification des fibrés holomorphes sur la sphère de Riemann. Amer. J. Math., 79:121–138, 1957.
hi:ch A. Hirschowitz, Le groupe de Chow équivariant, C. R. Acad. Sci. Paris Sér. I Math., 298(5):87–89, 1984.
hu:rg J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, Cambridge, 1990.
ki:co 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.
ki:qc Bumsig Kim, Quantum cohomology of flag manifolds ${G}/{B}$ and quantum Toda lattices, Ann. of Math. (2), 149(1):129–148, 1999.
ki:pf 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.
kl:gt S. L. Kleiman, The transversality of a general translate, Compositio Math., 28:287–297, 1974.
ko:ra J. Kollár, Rational curves on algebraic varieties, Springer-Verlag, Berlin, 1996.
ko:lo 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.
km:qh M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys., 164(3):525–562, 1994.
kt:lg A. Kresch and H. Tamvakis, Quantum cohomology of the Lagrangian Grassmannian, J. Algebraic Geom. 12 (2003), 777–810.
pe:mi D. Peterson, Lectures on quantum cohomology of G/P, M.I.T., 1996.
po:af Alexander Postnikov. Affine approach to quantum Schubert calculus. math.CO/ 0205165.
si:qc 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.
th:ir J. F. Thomsen, Irreducibility of $\overline {{M}}_ {0,n}({G}/{P},\beta )$, Internat. J. Math., 9(3):367–376, 1998.
wi:vl 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.
wo:pe C. Woodward. On D. Peterson’s comparison formula for Gromov-Witten invariants of ${G}/{P}$. math.AG/0206073.
yo:bo A. Yong, On bounds for quantum multiplication, Proc. Amer. Math. Soc. 131 (2003), 2649–2655.
ag:ei S. Agnihotri and C. Woodward, Eigenvalues of products of unitary matrices and quantum Schubert calculus, Math. Res. Lett., 5(6):817–836, 1998.
as:qc A. Astashkevich and V. Sadov, Quantum cohomology of partial flag manifolds ${F}_{n_ 1\cdots n_ k}$, Comm. Math. Phys., 170:503–528,1995.
bm:st K. Behrend and Yu. Manin, Stacks of stable maps and Gromov-Witten invariants, Duke Math. J., 1:1–60, 1996.
be:tr P. Belkale. Transformation formulas in Quantum Cohomology, 2001. preprint.
be:qs A. Bertram, Quantum schubert calculus, Adv. Math., 128:289–305, 1997,
be:qm A. Bertram, I. Ciocan-Fontanine, and W. Fulton, Quantum multiplication of Schur polynomials, J. Algebra, 219(2):728–746, 1999.
be:gi 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.
bo:lag A. Borel, Linear algebraic groups, Springer-Verlag, New York, second edition, 1991.
bo:rf R. Bott, A residue formula for holomorphic vector-fields, J. Differential Geometry, 1:311–330, 1967,
bo:lg 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.
bu:qc A. Buch, Quantum cohomology of Grassmannians, Compositio Math. 137 (2003), 227–235.
bkt:lg A. Buch, A. Kresch, and H. Tamvakis, Gromov-Witten invariants on Grassmannians, J. Amer. Math. Soc. 16 (2003), 901–915.
ca:de 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.
ch:su 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.
ch:qf L. Chen, Quantum cohomology of flag manifolds, Adv. Math. 174 (2003), 1–34.
cf:qc2 I. Ciocan-Fontanine, The quantum cohomology ring of flag varieties, Trans. Amer. Math. Soc., 351(7):2695–2729, 1999.
cf:qc3 I. Ciocan-Fontanine, On quantum cohomology rings of partial flag varieties, Duke Math. J., 98(3):485–524, 1999.
fgp:qs S. Fomin, S. Gelfand, and A. Postnikov, Quantum Schubert polynomials, J. Amer. Math. Soc., 10:565–596, 1997.
fu:ev W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc., 37:209–249, 2000.
fp:st 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.
gi:qc A. Givental and Bumsig Kim, Quantum cohomology of flag manifolds and Toda lattices, Comm. Math. Phys., 168(3):609–641, 1995.
gr:cl A. Grothendieck. Sur la classification des fibrés holomorphes sur la sphère de Riemann. Amer. J. Math., 79:121–138, 1957.
hi:ch A. Hirschowitz, Le groupe de Chow équivariant, C. R. Acad. Sci. Paris Sér. I Math., 298(5):87–89, 1984.
hu:rg J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, Cambridge, 1990.
ki:co 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.
ki:qc Bumsig Kim, Quantum cohomology of flag manifolds ${G}/{B}$ and quantum Toda lattices, Ann. of Math. (2), 149(1):129–148, 1999.
ki:pf 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.
kl:gt S. L. Kleiman, The transversality of a general translate, Compositio Math., 28:287–297, 1974.
ko:ra J. Kollár, Rational curves on algebraic varieties, Springer-Verlag, Berlin, 1996.
ko:lo 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.
km:qh M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys., 164(3):525–562, 1994.
kt:lg A. Kresch and H. Tamvakis, Quantum cohomology of the Lagrangian Grassmannian, J. Algebraic Geom. 12 (2003), 777–810.
pe:mi D. Peterson, Lectures on quantum cohomology of G/P, M.I.T., 1996.
po:af Alexander Postnikov. Affine approach to quantum Schubert calculus. math.CO/ 0205165.
si:qc 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.
th:ir J. F. Thomsen, Irreducibility of $\overline {{M}}_ {0,n}({G}/{P},\beta )$, Internat. J. Math., 9(3):367–376, 1998.
wi:vl 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.
wo:pe C. Woodward. On D. Peterson’s comparison formula for Gromov-Witten invariants of ${G}/{P}$. math.AG/0206073.
yo:bo 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
MR Author ID:
603893
Email:
ctw@math.rutgers.edu
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.
