Lexsegment and Gotzmann ideals associated with the diagonal action of Z/p

We consider a diagonal action of a cyclic group of prime order on a polynomial ring F[x (1), . . . , x (n) ]. We give a description of the actions for which the corresponding Hilbert ideal is Gotzmann when n = 2. Nevertheless, we show that there is a separating set of invariant monomials that generates a proper lexsegment ideal in the polynomial ring for all n. As well, we provide an algorithm to compute this set.