Varieties of Invariant Subspaces Given by Littlewood-Richardson Tableaux

Abstract

Given partitions $\alpha, \beta, \gamma$, the short exact sequences $0 \rightarrow N_\alpha \rightarrow N_\beta \rightarrow N_\gamma \rightarrow 0$ of nilpotent linear operators of Jordan types $\alpha, \beta, \gamma$, respectively, define a constructible subset $\mathbb{V}^\alpha_{\beta, \gamma}$ of an affine variety. Geometrically, the varieties $\mathbb{V}^\alpha_{\beta, \gamma}$ are of particular interest as they occur naturally and since they typically consist of several irreducible components. In fact, each Littlewood-Richardson tableaux $\Gamma$ of shape $(\alpha, \beta, \gamma)$ contributes one irreducible component $\overline{\mathbb{V}}_\Gamma$. We consider the partial order $\Gamma \leq^*_{closure} \tilde{\Gamma}$ on LR-tableaux which is the transitive closure of the relation given by $\mathbb{V}_{\tilde{\Gamma}} \cap \overline{\mathbb{V}_\Gamma} \neq 0$. In this paper we compare the closure-relation with partial orders given by algebraic, combinatorial and geometric conditions. In the case where the parts of $\alpha$ are at most two, all those partial orders are equivalent. We discuss how the orders differ in general.