The poset of the nilpotent commutator of a nilpotent matrix

Abstract

Let B be an n×n nilpotent matrix with entries in an infinite field k. The nilpotent commutator NB of B is known to be an irreducible algebraic variety. Assume that B is in Jordan canonical form with the associated Jordan block partition P and let Q(P) be the Jordan partition of a generic element of NB. P. Oblak associated to a given partition P another partition resulting from a recursive process. She conjectured that this partition is the same as Q(P). R. Basili, A. Iarrobino and the author later generalized the process introduced by Oblak. The main result of this paper states that all such processes give rise to the same partition – a partition that is defined in terms of the lengths of unions of special symmetric chains in a poset associated to NB. This poset was defined by P. Oblak as the digraph determined by the non-zero entries of a generic matrix in a maximal nilpotent subalgebra of NB.