Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-Intriligator formula

Discrete Mathematics

Volumn 217, Issue 1-3, 2000, Pages 191-223

  Kirillov, Anatol N ^a,b   Maeno, Toshiaki ^a  

^a UNIVERSITY OF TOKYO   (Japan)

^b STEKLOV MATHEMATICAL INSTITUTE   (Russian Federation)

Author keywords

Double quantum schubert polynomials; Ehresman Bruhat graph; Quantum Pieri's rule

Indexed keywords

EID: 0000836858     PISSN: 0012365X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0012-365X(99)00263-0     Document Type: Article

References (28)

1. A. Bertram, Quantum Schubert calculus, Adv. Math. 128 (1997) 289-305.
2. N. Bergeron, S. Billey, RC-graphs and Schubert polynomials, Exp. Math. 2 (1993) 257-269.
3. S. Billey, W. Jockusch, R. Stanley, Some combinatorial properties of Schubert polynomials, J. Algebraic Combin. 2 (1993) 345-374.
4. I. Ciocan-Fontanine I, Quantum cohomology of flag varieties, Int. Math. Res. Notes 6 (1995) 263-277.
5. I. Ciocan-Fontanine, Quantum cohomology ring of flag varieties, Preprint, 1997, 34p.
6. ∞ map germ, Ann. Math. 106 (1977) 19-38.
7. S. Fomin, S. Gelfand, A. Postnikov A, Quantum Schubert polynomials, J. Amer. Math. Soc. 10 (1997) 565-596.
8. S. Fomin, A.N. Kirillov, Quadratic algebras, Dunkl elements, and Schubert calculus, Preprint AMSPPS#Advances 199703-005-001, 1997, 34p.
9. W. Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J. 65 (1991) 381-420.
10. A. Givental, B. Kim, Quantum cohomology of flag manifolds and Toda lattices, Comm. Math. Phys. 168 (1995) 609-641.
11. Ph. Griffiths, J. Harris, Principles of Algebraic Geometry, Wiley, New York, 1978.
12. B. Kim, Quot schemes for flags and Gromov invariants for flag varieties, Preprint, 1995, alg-geom/9512003.
13. B. Kim, On equivariant quantum cohomology, Int. Math. Research Notes 17 (1996) 841-851.
14. B. Kim, Quantum cohomology of flag manifolds G/B and quantum Toda lattices, 1996, Preprint, alg-geom/9607001.
15. A.N. Kirillov, Quantum Schubert polynomials and quantum Schur functions, Preprint CRM 2452, 1997, 20p; q-alg/9701005.
16. A.N. Kirillov, Cauchy identities for universal Schubert polynomials, Preprint q-alg/9703047, 1997, 16p.
17. A.N. Kirillov, T. Maeno, Quantum Schubert polynomials and the Vafa-Intriligator formula, Preprint UTMS-96-41, 1996, 50p.
18. A. Kohnert, S. Veigneau, Using Schubert toolkit to compute with polynomials in several variables, in 8th International Conference on Formal Power Series and Algebraic Combinatorics, Univ. of Minnesota, 1996, pp. 283-293.
19. A. Lascoux, M.-P. Schützenberger, Polynômes de Schubert, C.R. Acad. Sc. Paris t.294 (1982) 447-450.
20. A. Lascoux, M.-P. Schützenberger, Symmetry and flag manifolds, Lecture Notes in Mathematics 996 (1983) 118-144.
21. I.G. Macdonald, Notes on Schubert polynomials, Publ. LCIM, Univ. de Quebec a Montreal, 1991.
22. I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd Edition, Oxford University Press, New York, 1995.
23. D. McDuff, D. Salamon, J-Holomorphic Curves and Quantum Cohomology, vol. 6, Univ. Lect., AMS; Providence, RI, 1994.
24. A. Postnikov, On Quantum Monk's and Pieri's Formulas, Preprint AMSPPS #199703-05-002, 1997, 12p.
25. Y. Ruan, G. Tian, Mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995) 259-367.
26. F. Sottile, Pieri's rule for flag manifolds and Schubert polynomials, Ann. Inst. Fourier 46 (1996) 89-110.
27. E. Witten, The Verlinde algebra and the cohomology of Grassmannian, Geometry, topology & physics, Conference Proceedings Lecture Notes, Geom. Topology, VI; Intern. Press, Cambridge MA, 1995, pp. 357-422.
28. E. Witten, The Verlinde algebra and the cohomology of Grassmannian, Geometry, topology & physics, Conference Proceedings Lecture Notes, Geom. Topology, VI; Intern. Press, Cambridge MA, 1995, pp. 357-422.