Let G be a linear, real reductive group, and let P be a parabolic subgroup. TheBruhat decomposition of G gives a cellular decomposition of the generalized flag manifold X = G/P. Originating in the work of Schubert on Grassmann manifolds, this cellular the decomposition was used the Ehresmann (Eh) to give a proof of the basis theorem for the integral cohomology of Grassmannians. Since Ehresmann it has been clear that, for a generalized flag manifold X, a more a detailed understanding of this decomposition by generalized Schubert cells would provide a determination of the integral (co-)homology of X. The aim of this article is to give a representation theoretic determination of the differentials in the Schubert cell decomposition of X and thereby obtain a combinatorial description of the integral (co-)homology of X. Our primary tool will be the infinite-dimensional representation theory of G.
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