Emilie Dufresne

Separating invariants 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.

The Cohen–Macaulay property of separating invariants of finite groups

In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply the ring of invariants is non-Cohen–Macaulay actually imply that no graded separating algebra is Cohen–Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded separating algebra is Cohen–Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen–Macaulay graded separating algebra implies the group is generated by bireections. Additionally, we give an example which shows that Cohen–Macaulay separating algebras can occur when the ring of invariants is not Cohen–Macaulay.

A Finite Separating Set for Daigle and Freudenburg's Counterexample to Hilbert's Fourteenth Problem

This article gives the first explicit example of a finite separating set in an invariant ring which is not finitely generated, namely, for Daigle and Freudenburgs 5-dimensional counterexample to Hilberts Fourteenth Problem.

Invariants and separating morphisms for algebraic group actions

The first part of this paper is a refinement of Winkelmann’s work on invariant rings and quotients of algebraic group actions on affine varieties, where we take a more geometric point of view. We show that the (algebraic) quotient

The separating variety for the basic representations of the additive group

X/\!/ G

Determinantal generalizations of instrumental variables

X//G given by the possibly not finitely generated ring of invariants is “almost” an algebraic variety, and that the quotient morphism

Sampling real algebraic varieties for topological data analysis

\pi :X \rightarrow X/\!/G