The combinatorial quantum cohomology ring of $G/B$

Abstract

A purely combinatorial construction of the quantum cohomology ring of the flag manifold $G/B$ is presented. We show that the ring we construct is commutative, associative and satisfies the usual grading condition. By using results of two of our previous papers, we obtain a presentation of this ring in terms of generators and relations, as well as formulas for quantum Giambelli polynomials. We show that these polynomials satisfy a certain orthogonality property, which - for G=SL_n(C) - was proved previously by Fomin, Gelfand and Postnikov.