Quantum cohomology of partial flag manifolds $$F_{n_1 } \ldots _{n_k }$$

AbstractWe compute the quantum cohomology rings of the partial flag manifolds $$F_{n_1 } \ldots _{n_k } = U(n)/(U(n_1 ) \times \cdots \times U(n_k ))$$ . The inductive computation uses the idea of Givental and Kim [1]. Also we define a notion of the vertical quantum cohomology ring of the algebraic bundle. For the flag bundle $$F_{n_1 } \ldots _{n_k }$$ (E) associated with the vector bundleE this ring is found.