Orbits of the Centralizer of a Linear Operator

TL;DR

This paper analyzes the orbit structure of the centralizer of a linear operator on complex vector spaces, providing a detailed classification and enumeration useful for solving systems of linear differential equations.

Contribution

It offers a complete description of the orbit lattice and a generating function for counting orbits, advancing understanding of the centralizer's action.

Findings

Describes the orbit structure of the centralizer group action.

Provides a lattice structure for the set of orbits.

Offers a generating function for counting orbits by dimension.

Abstract

We describe the orbit structure for the action of the centralizer group of a linear operator on a finite-dimensional complex vector space. The main application is to the classification of solutions to a system of first-order ODEs with constant coefficients. We completely describe the lattice structure on the set of orbits and provide a generating function for the number of orbits in each dimension.