A quantum type deformation of the cohomology ring of flag manifolds

Abstract

Let q_1, ..., q_n be some variables and set K:=Z[q_1, ..., q_n]/(q_1q_2...q_n). We show that there exists a K-bilinear product \star on H^*(F_n;Z)\otimes K which is uniquely determined by some quantum cohomology like properties (most importantly, a degree two relation involving the generators and an analogue of the flatness of the Dubrovin connection). Then we prove that \star satisfies the Frobenius property with respect to the Poincar\'e pairing of H^*(F_n;Z); this leads immediately to the orthogonality of the corresponding Schubert type polynomials. We also note that if we pick k\in {1,...,n} and we formally replace q_k by 0, the ring (H^*(F_n;Z)\otimes K,\star) becomes isomorphic to the usual small quantum cohomology ring of F_n, by an isomorphism which is described precisely.