The boundary of the irreducible components for invariant subspace varieties

Abstract

Given partitions $α$, $β$, $γ$, the short exact sequences $0\to N_α\to N_β\to N_γ\to 0$ of nilpotent linear operators of Jordan types $α$, $β$, $γ$, respectively, define a constructible subset $\mathbb V_{α,γ}^β$ of an affine variety. Geometrically, the varieties $\mathbb V_{α,γ}^β$ are of particular interest as they occur naturally and since they typically consist of several irreducible components. In fact, each Littlewood-Richardson (LR-) tableau $Γ$ of shape $(α,β,γ)$ contributes one irreducible component $\overline{\mathbb V}_Γ$. We consider the partial order $Γ\leq_{\sf bound}^*\widetildeΓ$ on LR-tableaux which is the transitive closure of the relation given by $\mathbb V_{\widetildeΓ}\cap \overline{\mathbb V}_Γ\neq \emptyset$. 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 $α$ are at most two, all those partial orders are equivalent. We prove that those partial orders are also equivalent in the case where $β\setminusγ$ is a horizontal and vertical strip. Moreover, we discuss how the orders differ in general.