How to predict whether an intersection of Schubert cycles will be a Schubert cycle?

I have read on "3264 and all that" the computation of the Chow ring of $G(1,3)$, the Grassmannian of lines in $\mathbb P^3$. So I'm trying to work out the Chow ring of $G=G(1,4)$, just as an exercise to get my hands dirty. But now I feel like they are too dirty. In fact, so far I determined the Schubert cycles, and those products in complementary codimension: this is OK. But then, I passed to the products where the sum of the codimensions is $5$. Now, $A^5G$ has one generator which is $\sigma_{32}$. So, please correct me if I'm wrong: when I intersect two Schubert cycles whose codimensions sum up to $5$, I have to get (either $0$ or) $\Sigma_{32}$ relative to some flag. Is this correct?

And when I'll move to $A^4G$, which is generated by two classes, there will be two possibilities: either I'll get a Schubert cycle, or I won't. So the question is: since it is not trivial (to me) to find the adequate flag in each case, is there at least a way to "predict" whether an intersection of Schubert cycles will be a Schubert cycle?

Thank you!

• Perhaps I wrote a silly thing (in brackets) when I wrote that one can get $0$. One always gets $\Sigma_{32}$ right? the intersection of two Schubert cycles whose codimensions sum up to $5$ cannot be empty... – Brenin Nov 18 '12 at 16:48