Multiplication in the cohomology of Grassmannians via Gröbner bases

In order to gain better understanding of the multiplication in the integral cohomology of the complex Grassmann manifold Gk,n(C) (in the Borel picture) a minimal strong Gröbner basis for the ideal Ik,n determining this cohomology is obtained. These results are applied to obtain recurrence relations among Kostka numbers which completely determine these numbers. Corresponding results for real Grassmann manifolds are also presented.