Separating invariants for the Klein four group and cyclic groups (Q2842044)

One of the main goals in invariant theory is to find generators of the invariant ring \(F[V]^G\), where \(V\) is a finite-dimensional representation of a finite group \(G\) over an algebraically closed field \(F\). A subset \(A\subset F[V]^G\) is said to be separating for \(V\) if for any pair of vectors \(u,w\in V\), we have: If \(f(u)=f(w)\) for all \(f\in A\), then \(f(u)=f(w)\) for all \(f\in F[V]^G\). In the study of separating invariants, the goal is to find a subalgebra of the ring of invariants which separates the group orbits. In this paper, the authors construct separating invariants for the indecomposable representations of the Klein four group over a field of characteristic 2, and of a cyclic group of order \(pm\) with \(p,m\) coprime over a field of characteristic \(p\). The separating sets consist of invariant polynomials that are almost exclusively orbit sums and products.