Regularity action of abelian linear groups on C^n

TL;DR

This paper characterizes the action of abelian subgroups of GL(n, C) on complex n-space, showing all orbits are regular with bounded order and providing methods to determine and classify these orbits.

Contribution

It offers a complete characterization of abelian linear group actions on C^n, including explicit methods for finitely generated groups and orbit isomorphism classification.

Findings

All orbits are regular with order ≤ 2n

Provided a method to determine the orbit order

Explicit characterization for finitely generated groups

Abstract

In this paper, we give a characterization of the action of any abelian subgroup G of GL(n, C) on C^n. We prove that any orbit of G is regular with order m<=2n. Moreover, we give a method to determine this order. In the other hand, we specify the region of all orbits which are isomorphic. If G is finitely generated, this characterization is explicit.