COMMUTING NILPOTENT MATRICES AND ARTINIAN ALGEBRAS

Fix an n x n nilpotent matrix B whose Jordan blocks are given by the partition P of n. Consider the ring C-B subset of Mat(n) (k) of n x n matrices with entries in an algebraically closed field k that commute with B, and its subset, the variety N-B subset of C-B of those that are nilpotent. Then N-B is an irreducible algebraic variety: so there is a Jordan block partition Q(P) of the generic matrix A is an element of N-B, that is greater than any other Jordan partition occurring for elements of N-B. What is Q(P)? We here introduce an algebra epsilon(B) whose radical is U-B, a maximal nilpotent subalgebra of N-B. We study the poset D-P, related to the digraph used by Oblak and Kosir [13]. Using our results, we give new, simpler proofs for much of what is known about Q(P), often clarifying or reducing the assumptions needed.