B-orbits of 2-nilpotent matrices and generalizations

TL;DR

This paper classifies B-orbits of 2-nilpotent matrices using oriented link patterns, extends Melnikov's work, and explores orbit closures and degenerations, also initiating a study on B-orbits of arbitrary nilpotent matrices.

Contribution

It introduces a classification of B-orbits on 2-nilpotent matrices via oriented link patterns and extends the analysis to arbitrary nilpotent matrices.

Findings

Classification of B-orbits on 2-nilpotent matrices

Description of orbit closures and minimal degenerations

Initiation of study on B-orbits of arbitrary nilpotent matrices

Abstract

The orbits of the group B of upper-triangular matrices acting on 2-nilpotent complex matrices via conjugation are classified via oriented link patterns, generalizing A. Melnikov's classification of the B-orbits on upper-triangular such matrices. The orbit closures as well as the "building blocks" of minimal degenerations of orbits are described. The classification uses the theory of representations of finite-dimensional algebras. Furthermore, we initiate the study of the B-orbits on arbitrary nilpotent matrices.