Invariant subspaces of nilpotent operators and LR-sequences

The aim of this paper is to study systematically invariant subspaces of finitedimensional nilpotent operators. Our main motivation comes from classifying the similarity orbit in thelattice of invariant subspaces of a given nilpotent operator. We give a detailed study of the Littlewood-Richardson similarity orbit. We show that none of the natural similarity relations is equivalent with the others.