The boundary of the irreducible components for invariant subspace varieties

TL;DR

This paper investigates the geometric and combinatorial boundaries of irreducible components in invariant subspace varieties related to nilpotent operators, establishing equivalences of partial orders in specific cases and exploring their differences.

Contribution

It introduces a new partial order on LR-tableaux and proves its equivalence with existing orders in cases where eta\setminus\gamma is a horizontal and vertical strip.

Findings

Partial orders on LR-tableaux are equivalent when parts of \alpha are at most two.

Equivalence of orders also holds when \beta\setminus\gamma is a horizontal and vertical strip.

The paper discusses how these orders differ in the general case.

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.