Quantum double Schubert polynomials represent Schubert classes

Abstract

The quantum double Schubert polynomials studied by Kirillov and Maeno, and by Ciocan-Fontanine and Fulton, are shown to represent Schubert classes in Kim’s presentation of the equivariant quantum cohomology of the flag variety. Parabolic analogues of quantum double Schubert polynomials are introduced and shown to represent Schubert classes in the equivariant quantum cohomology of partial flag varieties. This establishes a new method for computing equivariant Gromov-Witten invariants for partial flag varieties. For complete flags Anderson and Chen have announced a proof with different methods.