Degree bounds in quantum Schubert calculus

Yong, Alexander

doi:10.1090/S0002-9939-03-06850-3

Degree bounds in quantum Schubert calculus
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by Alexander Yong

Proc. Amer. Math. Soc. 131 (2003), 2649-2655

DOI: https://doi.org/10.1090/S0002-9939-03-06850-3

Published electronically: January 8, 2003

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Abstract:

Fulton and Woodward have recently identified the smallest degree of $q$ 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.

References

Aaron Bertram, Quantum Schubert calculus, Adv. Math. 128 (1997), no. 2, 289–305. MR 1454400, DOI 10.1006/aima.1997.1627
Aaron Bertram, Ionuţ Ciocan-Fontanine, and William Fulton, Quantum multiplication of Schur polynomials, J. Algebra 219 (1999), no. 2, 728–746. MR 1706853, DOI 10.1006/jabr.1999.7960
A. Buch, Quantum cohomology of Grassmannians, e-print math.AG/0106268.
A. Buch, Littlewood-Richardson Calculator, software available at http://www-math. mit.edu/˜abuch.
William Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 3, 209–249. MR 1754641, DOI 10.1090/S0273-0979-00-00865-X
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
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, DOI 10.1090/pspum/062.2/1492534
W. Fulton and C. Woodward, preprint, 2001.
G. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, 16. Addison-Wesley Co. Reading, Mass., 1981.
M. Kontsevich and Y. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Mirror Symmetry II, Amer. Math. Soc., Providence, RI, 1997, 607-653.
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
A. Postnikov, Affine approach to quantum Schubert calculus, e-print math. CO/0205165.
Y. Ruan and G. Tian, A mathematical theory of quantum cohomology, Math. Res. Lett. 1 (1994) 269-278.
F. Sottile, Rational Curves on Grassmannians: systems theory, reality, and transverality, to appear in Contemporary Mathematics, 2001.
Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282, DOI 10.1017/CBO9780511609589