Pairs of commuting nilpotent operators with one-dimensional intersection of kernels and matrices commuting with a Weyr matrix

TL;DR

This paper provides a simple, canonical normal form for pairs of commuting nilpotent matrices with a one-dimensional intersection of kernels, under specific conditions, simplifying classification in this complex problem.

Contribution

It introduces a new normal form for such matrix pairs, making classification more tractable when the Jordan matrix is a sum of equal-sized blocks over a zero characteristic field.

Findings

Normal form for pairs of commuting nilpotent matrices with one-dimensional kernel intersection

Canonical form established under specific Jordan block conditions

Simplifies classification problem in certain nilpotent operator cases

Abstract

I.M. Gelfand and V.A. Ponomarev (1969) proved that the problem of classifying pairs (A,B) of commuting nilpotent operators on a vector space contains the problem of classifying an arbitrary t-tuple of linear operators. Moreover, it contains the problem of classifying representations of an arbitrary quiver, and so it is considered as hopeless. We give a simple normal form of the matrices of (A,B) if the intersection of kernels of A and B is one-dimensional. We prove that this form is canonical if the Jordan matrix of A is a direct sum of Jordan blocks of the same size and the field is of zero characteristic.