Separating algebras and finite reflection groups

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.

12 pages, corrected yet another typo

References in corpus (1)

• Revêtements étales et groupe fondamental (SGA 1)

Cited by in corpus (11)

• Estimation under group actions: recovering orbits from invariants
• The Cohen-Macaulay property of separating invariants of finite groups
• Helly dimension of algebraic groups
• Separating invariants over finite fields
• Separating invariants for 2x2 matrices
• A finite separating set for Daigle and Freudenburg's counterexample to Hilbert's Fourteenth Problem
• Finite separating sets and quasi-affine quotients
• Degree bounds for fields of rational invariants of $\mathbb{Z}/p\mathbb{Z}$ and other finite groups
• The separating variety for the basic representations of the additive group
• Finite $F$-representation type for homogeneous coordinate rings of non-Fano varieties
• A rotation group whose subspace arrangement is not from a real reflection group