Constructively describing orbit spaces of finite groups by few inequalities

TL;DR

This paper provides explicit inequalities to describe the orbit spaces of finite groups acting on real space, improving understanding and simplifying the description of these spaces with elementary algebraic tools.

Contribution

It offers a new proof of Procesi and Schwarz's theorem and constructs explicit inequalities for abelian groups and cases needing only one inequality.

Findings

Explicit inequalities for abelian groups

Description of orbit spaces with minimal inequalities

Answer to Br"ocker's open question on genericity

Abstract

Let $G$ be a finite group acting linearly on $\mathbb{R}^n$. A celebrated Theorem of Procesi and Schwarz gives an explicit description of the orbit space $\mathbb{R}^n /\!/G$ as a basic closed semi-algebraic set. We give a new proof of this statement and another description as a basic closed semi-algebraic set using elementary tools from real algebraic geometry. Br\"ocker was able to show that the number of inequalities needed to describe the orbit space generically depends only on the group $G$. Here, we construct such inequalities explicitly for abelian groups and in the case where only one inequality is needed. Furthermore, we answer an open question raised by Br\"ocker concerning the genericity of his result.