Algebras of curvature forms on homogeneous manifolds

Let C(X) be the algebra generated by the curvature 2-forms of the standard hermitian line bundles over the complex homogeneous manifold X=G/B. We calculate the Hilbert polynomial of C(X) and give its presentation as a quotient of a polynomial ring. In particular, we show the dimension of C(X) is equal to the number of independent subsets of roots in the corresponding root system. As a tool we study a more general algebra associated with a point on a Grassmannian and calculate its Hilbert polynomial as well as its presentation in terms of generators and relations.