Vector fields invariant under a linear action of a compact Lie group

TL;DR

This paper proves that the module of smooth vector fields invariant under a compact Lie group's linear action on Euclidean space is finitely generated by polynomial invariant vector fields, linking smooth and polynomial invariants.

Contribution

It establishes that invariant smooth vector fields under a compact Lie group action are finitely generated by polynomial invariants, bridging smooth and algebraic invariant theory.

Findings

Invariant smooth vector fields are finitely generated by polynomial invariants.

The module of invariant vector fields is finitely generated.

Polynomial invariant vector fields generate all smooth invariant vector fields.

Abstract

This note shows that the module of smooth vector fields on ${\mathbb{R}}^n$, which are invariant under the linear action of a compact Lie group $G$ is finitely generated by polynomial vector fields on ${\mathbb{R}}^n$ which are invariant under the action of $G$.