Skip to Main Content

Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2024 MCQ for Journal of the American Mathematical Society is 4.83.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Gromov-Witten invariants on Grassmannians
HTML articles powered by AMS MathViewer

by Anders Skovsted Buch, Andrew Kresch and Harry Tamvakis;
J. Amer. Math. Soc. 16 (2003), 901-915
DOI: https://doi.org/10.1090/S0894-0347-03-00429-6
Published electronically: May 1, 2003

Abstract:

We prove that any three-point genus zero Gromov-Witten invariant on a type $A$ Grassmannian is equal to a classical intersection number on a two-step flag variety. We also give symplectic and orthogonal analogues of this result; in these cases the two-step flag variety is replaced by a sub-maximal isotropic Grassmannian. Our theorems are applied, in type $A$, to formulate a conjectural quantum Littlewood-Richardson rule, and in the other classical Lie types, to obtain new proofs of the main structure theorems for the quantum cohomology of Lagrangian and orthogonal Grassmannians.
References
  • L. Bégin, C. Cummins, and P. Mathieu, Generating-function method for fusion rules, J. Math. Phys. 41 (2000), no. 11, 7640–7674. MR 1788596, DOI 10.1063/1.1286512
  • L. Bégin, A. N. Kirillov, P. Mathieu, and M. A. Walton, Berenstein-Zelevinsky triangles, elementary couplings, and fusion rules, Lett. Math. Phys. 28 (1993), no. 4, 257–268. MR 1237577, DOI 10.1007/BF00761494
  • L. Bégin, P. Mathieu, and M. A. Walton, $\widehat {\mathrm {su}}(3)_k$ fusion coefficients, Modern Phys. Lett. A 7 (1992), no. 35, 3255–3265. MR 1191281, DOI 10.1142/S0217732392002640
  • Nantel Bergeron and Frank Sottile, Schubert polynomials, the Bruhat order, and the geometry of flag manifolds, Duke Math. J. 95 (1998), no. 2, 373–423. MR 1652021, DOI 10.1215/S0012-7094-98-09511-4
  • 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
  • [Bu1]Buch A. S. Buch : Quantum cohomology of Grassmannians, Compositio Math., to appear. [Bu2]qmonk A. S. Buch : A direct proof of the quantum version of Monk’s formula, Proc. Amer. Math. Soc., to appear. [BKT1]BKT A. S. Buch, A. Kresch, and H. Tamvakis : Grassmannians, two-step flags, and puzzles, in preparation. [BKT2]BKT2 A. S. Buch, A. Kresch, and H. Tamvakis : Quantum Pieri rules for isotropic Grassmannians, in preparation.
  • Sergey Fomin and Anatol N. Kirillov, Quadratic algebras, Dunkl elements, and Schubert calculus, Advances in geometry, Progr. Math., vol. 172, Birkhäuser Boston, Boston, MA, 1999, pp. 147–182. MR 1667680
  • 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
  • Howard Hiller and Brian Boe, Pieri formula for $\textrm {SO}_{2n+1}/\textrm {U}_n$ and $\textrm {Sp}_n/\textrm {U}_n$, Adv. in Math. 62 (1986), no. 1, 49–67. MR 859253, DOI 10.1016/0001-8708(86)90087-3
  • [K]Kn A. Knutson : Private communication. [KTW]KTW A. Knutson, T. Tao and C. Woodward : The honeycomb model of $GL(n)$ tensor products II: Puzzles determine facets of the Littlewood-Richardson cone, J. Amer. Math. Soc., to appear. [KT1]KTlg A. Kresch and H. Tamvakis : Quantum cohomology of the Lagrangian Grassmannian, J. Algebraic Geom., to appear. [KT2]KTorth A. Kresch and H. Tamvakis : Quantum cohomology of orthogonal Grassmannians, Compositio Math., to appear.
  • Piotr Pragacz, Algebro-geometric applications of Schur $S$- and $Q$-polynomials, Topics in invariant theory (Paris, 1989/1990) Lecture Notes in Math., vol. 1478, Springer, Berlin, 1991, pp. 130–191. MR 1180989, DOI 10.1007/BFb0083503
  • P. Pragacz and J. Ratajski, Formulas for Lagrangian and orthogonal degeneracy loci; $\~Q$-polynomial approach, Compositio Math. 107 (1997), no. 1, 11–87. MR 1457343, DOI 10.1023/A:1000182205320
  • Bernd Siebert and Gang Tian, On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator, Asian J. Math. 1 (1997), no. 4, 679–695. MR 1621570, DOI 10.4310/AJM.1997.v1.n4.a2
  • Frank Sottile, Pieri’s formula for flag manifolds and Schubert polynomials, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 1, 89–110 (English, with English and French summaries). MR 1385512, DOI 10.5802/aif.1508
  • [Tu]Tu G. Tudose : On the combinatorics of $sl(n)$-fusion coefficients, preprint (2001). [Y]Y A. Yong : Degree bounds in quantum Schubert calculus, Proc. Amer. Math. Soc., to appear.
Similar Articles
Bibliographic Information
  • Anders Skovsted Buch
  • Affiliation: Matematisk Institut, Aarhus Universitet, Ny Munkegade, 8000 Århus C, Denmark
  • MR Author ID: 607314
  • Email: abuch@imf.au.dk
  • Andrew Kresch
  • Affiliation: Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395
  • MR Author ID: 644754
  • Email: kresch@math.upenn.edu
  • Harry Tamvakis
  • Affiliation: Department of Mathematics, Brandeis University - MS 050, P. O. Box 9110, Waltham, Massachusetts 02454-9110
  • Email: harryt@brandeis.edu
  • Received by editor(s): July 18, 2002
  • Published electronically: May 1, 2003
  • Additional Notes: The authors were supported in part by NSF Grant DMS-0070479 (Buch), an NSF Postdoctoral Research Fellowship (Kresch), and NSF Grant DMS-0296023 (Tamvakis).
  • © Copyright 2003 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 16 (2003), 901-915
  • MSC (2000): Primary 14N35; Secondary 14M15, 14N15, 05E15
  • DOI: https://doi.org/10.1090/S0894-0347-03-00429-6
  • MathSciNet review: 1992829