Automorphisms and twisted forms of rings of invariants

TL;DR

This paper investigates the automorphism groups of rings of invariants under finite group actions, revealing that in many cases these groups are simply the multiplicative group of the base field, thus clarifying their structure.

Contribution

It provides new results on the automorphism groups of invariant rings, showing they are often just the scalar multiplications, and extends previous work with additional insights.

Findings

Automorphism group often equals the multiplicative group of the field

Results apply to many cases of finite subgroup actions

Extends prior research with new cases and methods

Abstract

Let $G\subset GL_n(k)$ be a finite subgroup and $k[x_1,\dots, x_n]^G\subset k[x_1,\dots, x_n]$ its ring of invariants. We show that, in many cases, the automorphism group of $k[x_1,\dots, x_n]^G$ is $k^\times$. Version 2: Incorporates parts of arXiv:2210.16265.