I need to compute the ring of cohomologies over the integers of the complex grassmannian G(4,2).
As I understand, one can use the Schubert cells and cellular homology to show that the homology groups of G(4,2) are free abelian with bases corresponding to the appropriate Schubert cells. And the cohomology groups have the same structure, am I right?
Now - I want to understand how the cup product looks like in this case without using some general formulas (like Pieri's or Giambelli's) which seem to be hard to prove. How can I do that? I don't understand how one can compute cup products using cellular cohomology. Thank you.