The Regular property of Invariant Rings over Regular Domains

TL;DR

This paper generalizes the Chevalley-Shephard-Todd theorem to invariant rings over regular domains, establishing a criterion for regularity based on the generation of the group by pseudo-reflections.

Contribution

It extends classical invariant theory results to regular domains, linking the regularity of invariant rings to the pseudo-reflection generation of the acting group.

Findings

Invariant ring regularity corresponds to the group being generated by pseudo-reflections.

The theorem applies to finite groups acting on polynomial rings over regular domains.

The result generalizes known theorems from fields to more general regular domains.

Abstract

The main result of this paper is a generalization of the theorem of Chevalley-Shephard-Todd to the rings of invariants of pseudo-reflection groups over regular domains. More precisely, let $A$ be a regular domain and let $K$ be its field of fractions. Let $G\subseteq GL_n(A)$ be a finite group. Let $G$ act linearly on $A[X_1,X_2,\dots, X_n]$ (fixing $A$). Assume that $|G|$ is invertible in $A$. We prove that $G\subseteq GL_n(K)$ is generated by pseudo-reflections if and only if $(A[X_1,X_2,\dots, X_n])^G$ is regular.