Annals of Combinatorics 4 (2000) 299-305


From Quantum Cohomology to Algebraic Combinatorics: The Example of Flag Manifolds

Rudolf Winkel

Institut für Reine und Angewandte Mathematik, RWTH Aachen, D-52056 Aachen, Germany
winkel@iram.rwth-aachen.de

Received February 16, 1999

AMS Subject Classification: 14M15, 05E15, 14N10

Abstract. The computation and understanding of quantum cohomology is a very hard problem in mathematical physics (string theory). We review in non-technical terms how, in the case of the flag manifolds, this problem can turn out to be at its core a non-trivial problem in algebraic combinatorics.

Keywords: quantum cohomology, flag manifold, Schubert poynomial, elementary symmetric polynomial, standard elementary monomial


References

1.  A. Borel, Sur la cohomologie des espaces fibré principaux et des espaces homogènes des groupes Lie compacts, Ann. Math. 57 (1953) 115–207.

2.  S. Fomin, S. Gelfand, and A. Postnikov, Quantum Schubert polynomials, J. Amer. Math. Soc. 10 (1997) 565–596.

3.  A. Givental and B. Kim, Quantum cohomology of flag manifolds and Toda lattices, Comm. Math. Phys. 168 (1995) 609–641.

4.  M. Kontsevich and Y. Manin, Gromov–Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994) 525–562.

5.  A. Lascoux, Polynômes de Schubert, Une approache historique, Discrete Math. 139 (1995) 303–317.

6.  A. Lascoux and M.P. Schützenberger, Fontorialité de polynômes de Schubert, Contemp. Math. 88 (1989) 585–598.

7.  I.G. Macdonald, Schubert polynomials, in: Surveys in Combinatorics, A.D. Kendwell, Ed., London Mathematical Society Lecture Notes Series, Vol. 166, Cambridge University Press, Cambridge, 1991, pp. 73–99.

8.  I.G. Macdonald, Notes on Schubert polynomials, Publications du L.A.C.I.M., Vol. 6, Université du Québec, Montréal, 1991.

9.  I.G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1979.

10.  D.R. Morrison, Mirror symmetry and rational curves on quintic threefolds: A guide for mathematicians, J. Amer. Math. Soc. 6 (1993) 223–247.

11.  D.R. Morrison, Mathematical aspects of mirror symmetry, alg-geom /9609021, (1996), 76 pp.

12.  Y. Ruan and G. Tian, A Mathematical theory of quantum cohomology, J. Diff. Geom. 42 (1995) 259–367.

13.  B.E. Sagan, The Symmetric Group, Wadsworth & BrooksCole, Pacific Grove, 1991.

14.  C. Vafa, Mirror transform and string theory, in: Geometry, Topology, and Physics for Roul Bott, S.-T. Yau, Ed., Cambridge International Press, Cambrigde, Massachusetts, 1994.

15.  R. Winkel, Recursive and combinatorial properties of Schubert polynomials, Sem. Loth. Comb. B38c (1996), 29 pp.

16.  R. Winkel, On the expansion of Schur and Schubert polynomials into standard elementary monomials, Adv. Math. 134 (1998) 46–89.

17.  R. Winkel, On algebraic and combinatorial properties of Schur and Schubert polynomials, Bayreuther Mathematische Schriften, 59 (2000), 255 pp.

18.  E. Witten, Topological sigma models, Comm. Math. Phys. 118 (1988) 411–449.

19.  E. Witten, Two-dimensional gravity and intersection theory on moduli spaces, Surveys in Diff. Geom. 1 (1991) 243–310.


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