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

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. Our main interest is 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.