$n \in {\ensuremath{\mathbb{N}}}$.

If $A$ is connected graded then one can define the notion of a Hilbert series.

\[def: hilbseries\] If $A$ is a connected graded algebra then its *Hilbert series* with respect to its [$\mathbb{N}$]{}-grading is $H_A(t):=\sum_{n \in {\ensuremath{\mathbb{N}}}}(\text{dim}_kA_n)t^n$.

From now on we will refer to connected graded algebras as c.g. algebras for brevity. When $A$ is [$\mathbb{N}$]{}-graded or a c.g.algebra we will assume that $G$ acts on $A$ by ${\ensuremath{\mathbb{N}}}$-graded algebra automorphisms. This means that in addition to the requirement of acting by algebra automorphisms, if $a \in A_n$ then $a^g \in A_n$. The group of ${\ensuremath{\mathbb{N}}}$-graded algebra automorphisms is denoted by $\text{Aut}_{{\ensuremath{\mathbb{N}}}-\text{alg}}(A)$.

If $A$ is ${\ensuremath{\mathbb{N}}}$-graded then we assume that it is generated by finitely many elements that are homogeneous with respect to its ${\ensuremath{\mathbb{N}}}$-grading. We do not, however, necessarily assume that the generators lie in degree 1. Our algebras will often be written as quotients of a free algebra $k\{x_0,\ldots,x_n\}$, which can also be described as the tensor algebra $T(V)$ over the vector space $V=kx_0 + \ldots + kx_n$. If we work with an algebra of the form $T(V)/I$ for some ideal $I$, then we will refer to the generators of $I$ as the *defining relations* 