Constructing G-algebras

In this article we define G-algebras, that is, graded algebras on which a reductive group G, acts as gradation preserving automorphisms. Starting from a finite dimensional G-module V and the polynomial ring C[V], it is shown how one constructs a sequence of projective varieties V-k such that each point of V-k corresponds to a graded algebra with the same decomposition up to degree k as a G-module. After some general theory, we apply this to the case that V is the n+1-dimensional permutation representation of Sn+1, the permutation group on n+1 letters.