1. Homologie effective
We describe the integral cohomology rings of the flag manifolds of types Bₙ, Dₙ, G₂ and F₄ in terms of their Schubert classes. The main tool is the divided difference operators of Bernstein-Gelfand-Gelfand and Demazure. As an application, we compute the Chow rings of the corresponding complex algebraic groups, recovering thereby the results of R. Marlin.
We use known results on the characteristic rank of the canonical –plane bundle over the oriented Grassmann manifold to compute the generators of the –cohomology groups for . Drawing from the similarities of these examples with the general description of the cohomology rings of we conjecture some predictions.