Quantum Schubert polynomials

Abstract

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 oa, called Schubert classes, form an additive basis in H* (Fln 2) . To relate the two descriptions, one would like to determine which elements of 2[xl, ... , 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 Schiitzenberger [22] came up with a combinatorial version of this theory (for the type A) by introducing remarkable polynomial representatives of the Schubert classes oa called Schubert polynomials and denoted Gw. Recently, motivated by ideas that came from the string theory [31, 30], mathematicians defined, for any Kahler algebraic manifold X, the (small) quantum cohomology ring QH* (X, 2), 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 , 2) 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 , 2) (0 Z[ql,..., qn-1], where the qi are formal variables (deformation parameters). The multiplicative structure of the quantum cohomology is however