The defining ideals of conjugacy classes of nilpotent matrices and a conjecture of Weyman

TL;DR

This paper develops methods to generate defining ideals for conjugacy class closures of nilpotent matrices, finds explicit generating sets for specific shapes, and provides a counterexample to Weyman's conjecture.

Contribution

It introduces new techniques for producing reduced generating sets of ideals associated with nilpotent conjugacy classes and challenges a conjecture by Weyman with a counterexample.

Findings

Explicit generating sets for particular nilpotent shapes.

Counterexample to Weyman's conjecture on ideal generators.

Comparison of new generating sets with existing ones.

Abstract

Tanisaki introduced generating sets for the defining ideals of the schematic intersections of the closure of conjugacy classes of nilpotent matrices with the set of diagonal matrices. These ideals are naturally labeled by integer partitions. Given such a partition $\lambda$, we define several methods to produce a reduced generating set for the associated ideal $I_{\lambda}$. For particular shapes we find nice generating sets. By comparing our sets with some generating sets of $I_{\lambda}$ arising from a work of Weyman, we find a counterexample to a related conjecture of Weyman.