An Algebraic Computation of the Height of the First Chern Class of the Canonical K-Plane Bundle over the Complex Grassmannian CG_(n,k)

We prove algebraically that the height of the first Chern class of the canonical k-plane bundle over the complex Grassmannian CG_(n,k) is k(n-k) in H^* ( CG_(n,k);Q) for k=2,3,4, and any positive integer n≥2k. We use the technique of R. E. Stong, of embedding the rational cohomology ring of the complex Grassmannian CG_(n,k) into the rational cohomology ring of the complete complex flag manifold F_C (⏟(1,…,1)┬n) .