I was reading chapter 6 in the book of Harris on Algebraic geometry and came to the following puzzle.
It seems to me that every Schubert cell in a Grassmanian is obtaining by cutting the Grassmanian by a certain plane in its Plucker embedding (hope this is correct, I got this idea from Harris' book)
Now, I was thinking always, that there is a whole theory of Littlwood Richardson coefficients that explains the intersection theory of Grassmanian and in particular explains what are intersection numbers of various cells.
My question is as follows. Why do we have this complicated theory of Littlwood Richardson coefficients? (is this because Schubert cells don't always have expected dimension ?) Naively, if all Schubert cells were obtained by intersecting Grassmanian transversally by some planes the only number that would pop up would be the degree of Grassmanian in its Plucker embedding.