Understanding the cohomology ring of the Grassmannian Some background first: I'm trying to understand the solution of some enumerative geometry problems, such as proving that a smooth cubic contains $27$ lines. I know that this becomes easier once one understands the cohomology of the Grassmannian.
I know that the Grassmannian can be given a CW-complex structure, but I don't understand how to compute the actual cohomology ring. I think that is the subject of Schubert calculus, and names like Pieri's or Giambelli's formulas often pop up. But I have also read elsewhere, such as in Hatcher's book Vector Bundles and K-Theory, that one can use Chern classes to describe the cohomology ring.
My question is, how are the two approaches related, and, most importantly, what is a comprehensive textbook on the subject?

The two texts I have most often been referred to for studying the cohomology of the Grassmannian are Young Tableaux by Fulton and Symmetric Functions, Schubert Polynomials, and Degeneracy Loci by Manivel. From my perspective, Manivel takes a fairly combinatorial perspective while Part III of Fulton gives an excellent treatment of this question from a more geometric perspective.
