Degree bounds in quantum Schubert calculus
Author:
Alexander Yong
Journal:
Proc. Amer. Math. Soc. 131 (2003), 26492655
MSC (1991):
Primary 14M15; Secondary 05E05, 14N10
Published electronically:
January 8, 2003
MathSciNet review:
1974319
Fulltext PDF Free Access
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Abstract: Fulton and Woodward have recently identified the smallest degree of that appears in the expansion of the product of two Schubert classes in the (small) quantum cohomology ring of a Grassmannian. We present a combinatorial proof of this result, and provide an alternative characterization of this smallest degree in terms of the rim hook formula for the quantum product.
 1.
Aaron
Bertram, Quantum Schubert calculus, Adv. Math.
128 (1997), no. 2, 289–305. MR 1454400
(98j:14067), http://dx.doi.org/10.1006/aima.1997.1627
 2.
Aaron
Bertram, Ionuţ
CiocanFontanine, and William
Fulton, Quantum multiplication of Schur polynomials, J.
Algebra 219 (1999), no. 2, 728–746. MR 1706853
(2000k:14042), http://dx.doi.org/10.1006/jabr.1999.7960
 3.
A. Buch, Quantum cohomology of Grassmannians, eprint math.AG/0106268.
 4.
A. Buch, LittlewoodRichardson Calculator, software available at http://wwwmath. mit.edu/~abuch.
 5.
William
Fulton, Eigenvalues, invariant factors,
highest weights, and Schubert calculus, Bull.
Amer. Math. Soc. (N.S.) 37 (2000), no. 3, 209–249 (electronic). MR 1754641
(2001g:15023), http://dx.doi.org/10.1090/S027309790000865X
 6.
William
Fulton, Young tableaux, London Mathematical Society Student
Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With
applications to representation theory and geometry. MR 1464693
(99f:05119)
 7.
W.
Fulton and R.
Pandharipande, Notes on stable maps and quantum cohomology,
Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math.,
vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 45–96.
MR
1492534 (98m:14025)
 8.
W. Fulton and C. Woodward, preprint, 2001.
 9.
G. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, 16. AddisonWesley Co. Reading, Mass., 1981.
 10.
M. Kontsevich and Y. Manin, GromovWitten classes, quantum cohomology, and enumerative geometry, Mirror Symmetry II, Amer. Math. Soc., Providence, RI, 1997, 607653.
 11.
I.
G. Macdonald, Symmetric functions and Hall polynomials, 2nd
ed., Oxford Mathematical Monographs, The Clarendon Press Oxford University
Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science
Publications. MR
1354144 (96h:05207)
 12.
A. Postnikov, Affine approach to quantum Schubert calculus, eprint math. CO/0205165.
 13.
Y. Ruan and G. Tian, A mathematical theory of quantum cohomology, Math. Res. Lett. 1 (1994) 269278.
 14.
F. Sottile, Rational Curves on Grassmannians: systems theory, reality, and transverality, to appear in Contemporary Mathematics, 2001.
 15.
Richard
P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge
Studies in Advanced Mathematics, vol. 62, Cambridge University Press,
Cambridge, 1999. With a foreword by GianCarlo Rota and appendix 1 by
Sergey Fomin. MR
1676282 (2000k:05026)
 1.
 A. Bertram, Quantum Schubert Calculus, Adv. Math. 128 (1997) 289305. MR 98j:14067
 2.
 A. Bertram, I. CiocanFontanine and W. Fulton, Quantum Multiplication of Schur polynomials, Journal of Algebra 219 (1999) 728746. MR 2000k:14042
 3.
 A. Buch, Quantum cohomology of Grassmannians, eprint math.AG/0106268.
 4.
 A. Buch, LittlewoodRichardson Calculator, software available at http://wwwmath. mit.edu/~abuch.
 5.
 W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 3, 209249. MR 2001g:15023
 6.
 W. Fulton, Young tableaux, Cambridge University Press, 1997. MR 99f:05119
 7.
 W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, Algebraic geometrySanta Cruz 1995, Amer. Math. Soc., Providence, RI, 1997, 4596. MR 98m:14025
 8.
 W. Fulton and C. Woodward, preprint, 2001.
 9.
 G. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, 16. AddisonWesley Co. Reading, Mass., 1981.
 10.
 M. Kontsevich and Y. Manin, GromovWitten classes, quantum cohomology, and enumerative geometry, Mirror Symmetry II, Amer. Math. Soc., Providence, RI, 1997, 607653.
 11.
 I. Macdonald, Symmetric functions and Hall polynomials, Second edition, Oxford University Press, 1995. MR 96h:05207
 12.
 A. Postnikov, Affine approach to quantum Schubert calculus, eprint math. CO/0205165.
 13.
 Y. Ruan and G. Tian, A mathematical theory of quantum cohomology, Math. Res. Lett. 1 (1994) 269278.
 14.
 F. Sottile, Rational Curves on Grassmannians: systems theory, reality, and transverality, to appear in Contemporary Mathematics, 2001.
 15.
 R. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, 1999. MR 2000k:05026
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Additional Information
Alexander Yong
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email:
ayong@umich.edu
DOI:
http://dx.doi.org/10.1090/S0002993903068503
PII:
S 00029939(03)068503
Keywords:
GromovWitten invariants,
quantum cohomology,
Grassmannian,
Schubert calculus
Received by editor(s):
December 14, 2001
Received by editor(s) in revised form:
April 2, 2002
Published electronically:
January 8, 2003
Communicated by:
John R. Stembridge
Article copyright:
© Copyright 2003 American Mathematical Society
