This paper studies separating invariants of finite groups acting on affine varieties through automorphisms. Several results, proved by Serre, Dufresne, Kac-Watanabe and Gordeev, and Jeffries and Dufresne exist that relate properties of the invariant ring or a separating subalgebra to properties of the group action. All these results are limited to the case of linear actions on vector spaces. The goal of this paper is to lift this restriction by extending these results to the case of (possibly) non-linear actions on affine varieties.Under mild assumptions on the variety and the group action, we prove that polynomial separating algebras can exist only for reflection groups. The benefit of this gain in generality is demonstrated by an application to the semigroup problem in multiplicative invariant theory.Then we show that separating algebras which are complete intersections in a certain codimension can exist only for 2-reflection groups. Finally we prove that a separating set of size n + k − 1 (where n is the dimension of X) can exist only for k-reflection groups.Several examples show that most of the assumptions on the group action and the variety that we make cannot be dropped.
This note provides a set of separating invariants for the ring of vector invariants K[V 2 ] Sn of two copies of the natural Sn-representation V = K n over a field of characteristic 0. This set is much smaller than generating sets of K[V 2 ] Sn .For n ≤ 4 we show that this set is minimal with respect to inclusion among all separating sets.
This note considers a finite algebraic group G acting on an affine variety X by automorphisms. Results of Dufresne on polynomial separating algebras for linear representations of G are extended to this situation. For that purpose, we show that the Cohen-Macaulay defect of a certain ring is greater than or equal to the minimal number k such that the group is generated by (k + 1)-reflections. Under certain rather mild assumptions on X and G we deduce that a separating set of invariants of the smallest possible size n = dim(X) can exist only for reflection groups.
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