Pairs of commuting nilpotent matrices, and Hilbert function

TL;DR

This paper explores the structure of pairs of commuting nilpotent matrices over an infinite field, linking their properties to the Hilbert function of associated Artinian rings and classifying stable partitions.

Contribution

It establishes a connection between the Hilbert function of K[A,B] and the Jordan form of generic linear combinations, characterizes stable partitions, and extends known results in characteristic zero.

Findings

The generic element of A+λB has Jordan form with maximum partition P(H).

Stable partitions are those with parts differing by at least two.

Q(P) is shown to have decreasing parts and be stable.

Abstract

Let K be an infinite field and denote by H(n,K) the family of pairs (A,B) of commuting nilpotent n by n matrices with entries in K. There has been substantial recent study of the connection between H(n,K) and the fibre H[n] of the punctual Hilbert scheme of the plane, over an n-fold point of the symmetric product, by V. Baranovsky, R. Basili, and A. Premet. We study the stratification of H(n,K) by the Hilbert function of the Artinian ring K[A,B]. We show that when dim_K K[A,B] = n, then the generic element of the pencil A+\lambda B, \lambda \in K, has Jordan partition the maximum partition P(H) whose diagonal lengths are the Hilbert function of K[A,B]. We denote by Q(P) the maximum Jordan partition of a nilpotent A commuting with a nilpotent B of Jordan partition P. We show that the stable partitions - those such that Q(P)=P - are those whose parts differ by at least two. In characteristic zero, the latter is a special case of a result of D. Panyushev. Our result on pencils shows that Q(P) has decreasing parts. In related work, T. Kosir and P. Oblak have shown further that Q(P) is itself stable.