Box moves on Littlewood-Richardson tableaux and an application to invariant subspace varieties

TL;DR

This paper extends combinatorial methods using Littlewood-Richardson tableaux to classify embeddings in finite abelian groups and applies these results to analyze invariant subspace varieties of nilpotent operators.

Contribution

It introduces a new combinatorial approach to classify embeddings in finite abelian groups and applies it to invariant subspace varieties of nilpotent linear operators.

Findings

Generalized Kaplansky's classification to finite direct sums

Developed a criterion for boundary partial order of irreducible components

Connected combinatorial classification with invariant subspace analysis

Abstract

In his 1951 book "Infinite Abelian Groups", Kaplansky gives a combinatorial characterization of the isomorphism types of embeddings of a cyclic subgroup in a finite abelian group. In this paper we first use partial maps on Littlewood-Richardson tableaux to generalize this result to finite direct sums of such embeddings. We then focus on an application to invariant subspaces of nilpotent linear operators. We develop a criterion to decide if two irreducible components in the representation space are in the boundary partial order.