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 $\mathbb{C}[V]$, it is shown how one constructs a sequence of projective varieties $\mathbf{V}_k$ such that each point of $\mathbf{V}_k$ 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 $S_{n+1}$, the permutation group on $n+1$ letters.