Given a non-semistable vector bundle on $P^2$ of rank 4, are explicit conditions known for when the ranks in its Harder-Narasimhan filtration are (3,1), (2,2) and (1,3) respectively?
I would be very happy, if this is worked out somewhere, to be pointed to a reference.
I can see the resemblance of my question to this other question on moduli spaces of vector bundles with fixed HN filtration factors, but I am hoping something more concrete is known, perhaps in the low rank situations.
