Separating algebras and finite reflection groups

TL;DR

This paper introduces a geometric concept of separating algebras and proves that only groups generated by reflections or bireflections can have polynomial or complete intersection separating algebras, respectively.

Contribution

It establishes a geometric framework for separating algebras and characterizes groups with polynomial or complete intersection separating algebras.

Findings

Only groups generated by reflections have polynomial separating algebras.

Only groups generated by bireflections have complete intersection separating algebras.

Introduces a geometric notion of separating algebra.

Abstract

A separating algebra is, roughly speaking, a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants. In this paper, we introduce a geometric notion of separating algebra. This allows us to prove that only groups generated by reflections may have polynomial separating algebras, and only groups generated by bireflections may have complete intersection separating algebras.