Diagonal Orbits in a Type A Double Flag Variety of Complexity One

We continue our study of the inclusion posets of diagonal $SL(n)$-orbit closures in a product of two partial flag varieties. We prove that, if the diagonal action is of complexity one, then the poset is isomorphic to one of the 28 posets that we determine explicitly. Furthermore, our computations show that the number of diagonal $SL(n)$-orbits in any of these posets is at most 10 for any positive integer $n$. This is in contrast with the complexity 0 case, where, in some cases, the resulting posets attain arbitrary heights.