Quantum Schubert Polynomials Sergey

where In is the ideal generated by symmetric polynomials in x1, . . . , xn without constant term. Another, geometric, description of the cohomology ring of the flag manifold is based on the decomposition of Fln into Schubert cells. These are even-dimensional cells indexed by the elements w of the symmetric group Sn . The corresponding cohomology classes σw , called Schubert classes, form an additive basis in H ∗(Fln ,Z). To relate the two descriptions, one would like to determine which elements of Z[x1, . . . , xn]/In correspond to the Schubert classes under the isomorphism (1.1). This was first done in [2] (see also [8]) for a general case of an arbitrary complex semisimple Lie group. Later, Lascoux and Schützenberger [22] came up with a combinatorial version of this theory (for the type A) by introducing remarkable polynomial representatives of the Schubert classes σw called Schubert polynomials and denoted Sw . Recently, motivated by ideas that came from the string theory [31, 30], mathematicians defined, for any Kähler algebraic manifold X , the (small) quantum cohomology ring QH∗(X,Z), which is a certain deformation of the classical cohomology ring (see, e.g., [28, 19, 14] and references therein). The additive structure of QH∗(X ,Z) is essentially the same as that of ordinary cohomology. In particular, QH∗(Fln ,Z) is canonically isomorphic, as an abelian group, to the tensor product H∗(Fln ,Z) ⊗ Z[q1, . . . , qn−1], where the qi are formal variables (deformation parameters). The multiplicative structure of the quantum cohomology is however