The boundary of the irreducible components for invariant subspace varieties

Abstract

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 $\Gamma$ of shape $(\alpha,\beta,\gamma)$ contributes one irreducible component $\overline{\mathbb V}_\Gamma$. We consider the partial order $\Gamma\leq_{\sf bound}^*\widetilde{\Gamma}$ 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.