Pairs of commuting nilpotent operators with one-dimensional intersection of kernels and matrices commuting with a Weyr matrix
Representation Theory
2020-12-29 v1
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.
Keywords
Cite
@article{arxiv.2012.04038,
title = {Pairs of commuting nilpotent operators with one-dimensional intersection of kernels and matrices commuting with a Weyr matrix},
author = {Vitalij M. Bondarenko and Vyacheslav Futorny and Anatolii P. Petravchuk and Vladimir V. Sergeichuk},
journal= {arXiv preprint arXiv:2012.04038},
year = {2020}
}