On pairs of commuting nilpotent matrices

Abstract

Let B be a nilpotent matrix and suppose that its Jordan canonical form is determined by a partition λ. Then it is known that its nilpotent commutator \( \mathcal{N}_B\) is an irreducible variety and that there is a unique partition μ such that the intersection of the orbit of nilpotent matrices corresponding to μ with \( \mathcal{N}_{B}\) is dense in \( \mathcal{N}_{B}\). We prove that map \( \mathcal{D} \) given by \( \mathcal{D}{\left( \lambda \right)} = \mu \) is an idempotent map. This answers a question of Basili and Iarrobino [9] and gives a partial answer to a question of Panyushev [18]. In the proof, we use the fact that for a generic matrix \( A \in \mathcal{N}_{B}\) the algebra generated by A and B is a Gorenstein algebra. Thus, a generic pair of commuting nilpotent matrices generates a Gorenstein algebra. We also describe \( \mathcal{D}{\left( \lambda \right)} \) in terms of λ if \( \mathcal{D}{\left( \lambda \right)} \) has at most two parts.