A note on quantum products of Schubert classes in a Grassmannian
Dave Anderson
University of Michigan Department of Mathematics Ann Arbor MI 48109 USA Ann Arbor MI 48109 USA
DOI: 10.1007/s10801-006-0040-5
Abstract
Given two Schubert classes σ λ and σ μ in the quantum cohomology of a Grassmannian, we construct a partition ν , depending on λ and μ , such that σ ν appears with coefficient 1 in the lowest (or highest) degree part of the quantum product σ λ \bigstar σ μ . To do this, we show that for any two partitions λ and μ , contained in a k \times ( n - k) rectangle and such that the 180 \deg -rotation of one does not overlap the other, there is a third partition ν , also contained in the rectangle, such that the Littlewood-Richardson number c λ μ ν is 1.
Pages: 349–356
Keywords: keywords quantum cohomology; toric tableau; Littlewood-Richardson number
Full Text: PDF
References
1. A. Bertram, I. Ciocan-Fontanine, and W. Fulton, “Quantum multiplication of Schur polynomials,” J. Algebra 219 (1999), 728-746.
2. W. Fulton, Young Tableaux, Cambridge Univ. Press, 1997.
3. W. Fulton and C. Woodward, “On the quantum product of Schubert classes,” J. Algebraic Geom. 13 (2004), 641-661.
4. S. Kwon, “Real aspects of the moduli space of genus zero stable maps and real version of the Gromov-Witten theory,” math.AG/0305128.
5. A. Postnikov, “Affine approach to quantum Schubert calculus,” Duke Math. J. 128(3) (2005), 473-509.
6. R.P. Stanley, Enumerative Combinatorics, Volume 2, with appendage by S. Fomin, Cambridge, 1999.
7. M.P. Sch\ddot utzenberger, “La correspondance de Robinson,” in Combinatoire et Represéntation du Groupe Symétrique, Lecture Notes in Math., 579 (1977), Springer-Verlag, 59-135.
8. A. Yong, “Degree bounds in quantum Schubert calculus,” Proc. Amer. Math. Soc. 131(9) (2003), 2649-2655. 2 The phrase in quotes should be interpreted as follows: Let M = M0,3(X, d) be the Kontsevich moduli space of stable maps, and let M(R) be its real part. The Gromov-Witten invariants cνλμ(d) are certain intersection numbers in H * (M, Z); let cνλμ(d) be the analogous intersection numbers in H* (M(R), Z/2Z). It is reasonable to expect that cνλμ(d) \equiv cνλμ(d) (mod 2), as is true for the classical case (d = 0). An outline discussion of intersection theory on M(R) can be found in [4]. 3 Available at http://www.math.rutgers.edu/~asbuch/lrcalc/.
2. W. Fulton, Young Tableaux, Cambridge Univ. Press, 1997.
3. W. Fulton and C. Woodward, “On the quantum product of Schubert classes,” J. Algebraic Geom. 13 (2004), 641-661.
4. S. Kwon, “Real aspects of the moduli space of genus zero stable maps and real version of the Gromov-Witten theory,” math.AG/0305128.
5. A. Postnikov, “Affine approach to quantum Schubert calculus,” Duke Math. J. 128(3) (2005), 473-509.
6. R.P. Stanley, Enumerative Combinatorics, Volume 2, with appendage by S. Fomin, Cambridge, 1999.
7. M.P. Sch\ddot utzenberger, “La correspondance de Robinson,” in Combinatoire et Represéntation du Groupe Symétrique, Lecture Notes in Math., 579 (1977), Springer-Verlag, 59-135.
8. A. Yong, “Degree bounds in quantum Schubert calculus,” Proc. Amer. Math. Soc. 131(9) (2003), 2649-2655. 2 The phrase in quotes should be interpreted as follows: Let M = M0,3(X, d) be the Kontsevich moduli space of stable maps, and let M(R) be its real part. The Gromov-Witten invariants cνλμ(d) are certain intersection numbers in H * (M, Z); let cνλμ(d) be the analogous intersection numbers in H* (M(R), Z/2Z). It is reasonable to expect that cνλμ(d) \equiv cνλμ(d) (mod 2), as is true for the classical case (d = 0). An outline discussion of intersection theory on M(R) can be found in [4]. 3 Available at http://www.math.rutgers.edu/~asbuch/lrcalc/.