The boundary of the irreducible components for invariant subspace varieties

Given partitions $\alpha$, $\beta$, $\gamma$, the short exact sequences $0\to N_\alpha \to N_\beta \to N_\gamma \to 0$ of nilpotent linear operators of Jordan types $\alpha$, $\beta$, $\gamma$, respectively, define a constructible subset ${\mathbb{V}}_{\alpha ,\gamma }^{\beta }$ of an affine variety. Geometrically, the varieties ${\mathbb{V}}_{\alpha ,\gamma }^{\beta }$ are of particular interest as they occur naturally and since they typically consist of several irreducible components. In fact, each Littlewood-Richardson (LR-) tableau $\mathrm{\Gamma }$ of shape $(\alpha ,\beta ,\gamma )$ contributes one irreducible component ${\bar{\mathbb{V}}}_{\mathrm{\Gamma }}$. We consider the partial order $\mathrm{\Gamma }{\le }_{\mathsf{b}\mathsf{o}\mathsf{u}\mathsf{n}\mathsf{d}}^{\ast }\tilde{\mathrm{\Gamma }}$ on LR-tableaux which is the transitive closure of the relation given by $\mathbb V_{\widetilde{\Gamma}}\cap \overline{\mathbb V}_\Gamma\neq \emptyset$. 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 $\beta \setminus \gamma$ is a horizontal and vertical strip. Moreover, we discuss how the orders differ in general.