Geodesic
Gromov-Witten invariants on Grassmannians
Buch, Anders1 ; Kresch, Andrew2 ; Tamvakis, Harry3
1 Matematisk Institut, Aarhus Universitet, Ny Munkegade, 8000 Århus C, Denmark
2 Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395
3 Department of Mathematics, Brandeis University - MS 050, P. O. Box 9110, Waltham, Massachusetts 02454-9110
Journal of the American Mathematical Society, Tome 16 (2003) no. 4, pp. 901-915

Voir la notice de l'article provenant de la source American Mathematical Society

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.
DOI : 10.1090/S0894-0347-03-00429-6

Buch, Anders 1 ; Kresch, Andrew 2 ; Tamvakis, Harry 3

1 Matematisk Institut, Aarhus Universitet, Ny Munkegade, 8000 Århus C, Denmark
2 Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395
3 Department of Mathematics, Brandeis University - MS 050, P. O. Box 9110, Waltham, Massachusetts 02454-9110 
@article{10_1090_S0894_0347_03_00429_6,
     author = {Buch, Anders and Kresch, Andrew and Tamvakis, Harry},
     title = {Gromov-Witten invariants on {Grassmannians}},
     journal = {Journal of the American Mathematical Society},
     pages = {901--915},
     publisher = {mathdoc},
     volume = {16},
     number = {4},
     year = {2003},
     doi = {10.1090/S0894-0347-03-00429-6},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-03-00429-6/}
}
TY  - JOUR
AU  - Buch, Anders
AU  - Kresch, Andrew
AU  - Tamvakis, Harry
TI  - Gromov-Witten invariants on Grassmannians
JO  - Journal of the American Mathematical Society
PY  - 2003
SP  - 901
EP  - 915
VL  - 16
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-03-00429-6/
DO  - 10.1090/S0894-0347-03-00429-6
ID  - 10_1090_S0894_0347_03_00429_6
ER  - 
%0 Journal Article
%A Buch, Anders
%A Kresch, Andrew
%A Tamvakis, Harry
%T Gromov-Witten invariants on Grassmannians
%J Journal of the American Mathematical Society
%D 2003
%P 901-915
%V 16
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-03-00429-6/
%R 10.1090/S0894-0347-03-00429-6
%F 10_1090_S0894_0347_03_00429_6
Buch, Anders; Kresch, Andrew; Tamvakis, Harry. Gromov-Witten invariants on Grassmannians. Journal of the American Mathematical Society, Tome 16 (2003) no. 4, pp. 901-915. doi: 10.1090/S0894-0347-03-00429-6

[1] Bã©Gin, L., Cummins, C., Mathieu, P. Generating-function method for fusion rules J. Math. Phys. 2000 7640 7674

[2] Bã©Gin, L., Kirillov, A. N., Mathieu, P., Walton, M. A. Berenstein-Zelevinsky triangles, elementary couplings, and fusion rules Lett. Math. Phys. 1993 257 268

[3] Bã©Gin, L., Mathieu, P., Walton, M. A. ̂𝑠𝑢(3)_{𝑘} fusion coefficients Modern Phys. Lett. A 1992 3255 3265

[4] Bergeron, Nantel, Sottile, Frank Schubert polynomials, the Bruhat order, and the geometry of flag manifolds Duke Math. J. 1998 373 423

[5] Bertram, Aaron Quantum Schubert calculus Adv. Math. 1997 289 305

[6] Bertram, Aaron, Ciocan-Fontanine, Ionuå£, Fulton, William Quantum multiplication of Schur polynomials J. Algebra 1999 728 746

[7] Fomin, Sergey, Kirillov, Anatol N. Quadratic algebras, Dunkl elements, and Schubert calculus 1999 147 182

[8] Fulton, William Young tableaux 1997

[9] Fulton, W., Pandharipande, R. Notes on stable maps and quantum cohomology 1997 45 96

[10] Hiller, Howard, Boe, Brian Pieri formula for 𝑆𝑂_{2𝑛+1}/𝑈_{𝑛} and 𝑆𝑝_{𝑛}/𝑈_{𝑛} Adv. in Math. 1986 49 67

[11] Pragacz, Piotr Algebro-geometric applications of Schur 𝑆- and 𝑄-polynomials 1991 130 191

[12] Pragacz, P., Ratajski, J. Formulas for Lagrangian and orthogonal degeneracy loci Compositio Math. 1997 11 87

[13] Siebert, Bernd, Tian, Gang On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator Asian J. Math. 1997 679 695

[14] Sottile, Frank Pieri’s formula for flag manifolds and Schubert polynomials Ann. Inst. Fourier (Grenoble) 1996 89 110

Cité par Sources :