The boundary of the irreducible components for invariant subspace varieties

Given partitions alpha, beta, gamma, the short exact sequences 0 -> N-alpha -> N-beta -> N-gamma -> 0 of nilpotent linear operators of Jordan types alpha, beta, gamma, respectively, define a constructible subset V-alpha,gamma(beta) of an affine variety. Geometrically, the varieties V-alpha,gamma(beta) are of particular interest as they occur naturally and since they typically consist of several irreducible components. In fact, each Littlewood-Richardson tableau Gamma of shape (alpha, beta, gamma) contributes one irreducible component (V) over bar (Gamma). We consider the partial order Gamma <=(boundary) (Gamma) over tilde on LR-tableaux which is the transitive closure of the relation given by V (Gamma) over tilde boolean AND (V) over bar (Gamma) not equal empty set. In this paper we compare the boundary relation with partial orders given by algebraic, combinatorial and geometric conditions. It is known that in the case where the parts of alpha are at most two, all those partial orders are equivalent. We prove that those partial orders are also equivalent in the case where beta\gamma is a horizontal and vertical strip. Moreover, we discuss how the orders differ in general.