The Combinatorial Quantum Cohomology Ring of G/ B
Augustin-Liviu Mare
Department of Mathematics and Statistics University of Regina College West 307.14 Regina Saskatchewan Canada S4S.0A2
DOI: 10.1007/s10801-005-6915-z
Abstract
A purely combinatorial construction of the quantum cohomology ring of the generalized flag manifold is presented. We show that the ring we construct is commutative, associative and satisfies the usual grading condition. By using results of our previous papers [12, 13], we obtain a presentation of this ring in terms of generators and relations, and formulas for quantum Giambelli polynomials. We show that these polynomials satisfy a certain orthogonality property, which-for G = SL n( C {\cal C} )-was proved previously in the paper [5].
Pages: 331–349
Keywords: key words generalized flag manifolds; quantum cohomology; quantum Chevalley formula; quantum giambelli problem
Full Text: PDF
References
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12. A.-L. Mare “Polynomial representatives of Schubert classes in Q H * (G/B),” Math. Res. Lett. 9 (2002), 757-770.
13. A.-L. Mare “Relations in the quantum cohomology ring of G/B,” Math. Res. Lett. 11 (2004), 35-48.
14. A.-L. Mare “Quantum cohomology of the infinite dimensional generalized flag manifolds,” Adv. Math. 185 (2004), 347-369.
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2. A. Borel, “Sur la cohomologie des espaces fibrés principaux et des espaces homog`enes des groupes de Lie compacts,” Ann. of Math. (2) 57 (1953), 115-207.
3. F. Brenti, S. Fomin, and A. Postnikov, “Mixed Bruhat operators and Yang-Baxter equations for Weyl groups,” IMRN 8 (1999), 420-441.
4. C. Chevalley, “Invariants of finite groups generated by reflections,” Amer. J. Math. 77 (1955), 778-782.
5. S. Fomin, S. Gelfand, and A. Postnikov, “Quantum Schubert polynomials,” J. Amer. Math. Soc. 10 (1997), 565-596.
6. W. Fulton and R. Pandharipande, “Notes on stable maps and quantum cohomology,” in Algebraic geometry- Santa Cruz 1995, Proc. Sympos. Pure Math. 62, (Part 2), J. Kollar, R. Lazarsfeld and D.R. Morrison (eds.), 1997, pp. 45-96.
7. W. Fulton and C. Woodward, “On the quantum product of Schubert classes,” preprint math.AG/0112183.
8. R. Goodman and N.R. Wallach, “Classical and quantum-mechanical systems of Toda lattice type, I,” Comm. Math. Phys. 83 (1982), 355-386.
9. H. Hiller, Geometry of Coxeter Groups, Pitman Advanced Publishing Program, 1982.
10. J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, 1990.
11. B. Kim, “Quantum cohomology of flag manifolds G/B and quantum Toda lattices,” Ann. of Math. 149 (1999), 129-148.
12. A.-L. Mare “Polynomial representatives of Schubert classes in Q H * (G/B),” Math. Res. Lett. 9 (2002), 757-770.
13. A.-L. Mare “Relations in the quantum cohomology ring of G/B,” Math. Res. Lett. 11 (2004), 35-48.
14. A.-L. Mare “Quantum cohomology of the infinite dimensional generalized flag manifolds,” Adv. Math. 185 (2004), 347-369.
15. D. Peterson, “Lectures on quantum cohomology of G/P,” M.I.T., 1996.
16. A. Postnikov, Enumeration in Algebra and Geometry, Ph.D. thesis, M.I.T., 1997.