Horn conjecture is to show the sufficiency of the Horn inequalities. In our approach this amounts to showing that a non-transverse intersection on $Gr(r,{\mathbb{C}}^n)$ lifts to something upstairs which is not only non-transverse, but non-transverse for the aforementioned trivial reasons. However, once we have all the appropriate machinery in place, this turns out to be almost as straightforward as the “easy” direction of Horn’s conjecture.

The two-step flag manifolds are not a necessary part of the argument. In principle the entire argument could be formulated inside the tangent space of the Grassmannian. This would probably even lead to a shorter proof. However, it is our opinion that the geometry of the two-step flag manifold plays a fundamental role in this picture, and to undermine its role would be remiss.

Although there are a number of new ideas in this paper, we do not claim to be presenting an original or independent proof of the Horn conjecture. Our objective in writing this was to better understand the argument in [@B], to simplify it in places, and to show how it relates to some of our own previous work [@P1]. A few of the results have been taken directly from [@B], while 