2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/161371Let $B$ 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 $N_B$ 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 $N_B$ is dense in $N_B$. We prove that map $D$ given by $D(λ)=μ$ is an idempotent map. This answers a question of Basili and Iarrobino and gives a partial answer to a question of Panyushev. In the proof, we use the fact that for a generic matrix $A \in N_B$ 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 $D(λ)$ in terms of $λ$ if $D(λ)$ has at most two parts.7 pages, 1 figure, small changes, added motivation and referencesCommutative AlgebraAlgebraic Geometry13E10, 15A27text