How does Molien series describe polynomial invariants? As I understood from wiki page, 
Given a finite group acting on a vector space, 
Molien series gives a generating function, although I am not sure what this means. And how is this related to the polyomial invariants of the group? Could anyone help me here?
 A: The wikipedia page turned out to be quite useful, but also brief and a little faulty in some places, so I will rephrase this a little and correct wikipedia later.
Let $G$ be a finite group acting on a finite dimensional complex vector space $V\cong\mathbb{C}^n$ via $\rho:G\to\operatorname{GL}(V)$. We choose a basis $\vec{x}_1,\ldots,\vec{x}_n\in V$ and denote by $x_1,\ldots,x_n\in V^\ast$ the dual basis. Then, the polynomial functions on $V$ are quite literally the polynomial ring $\mathbb C[V]=\mathbb C[x_1,\ldots,x_n]$. The homogeneous polynomials of degree $d$ are $\mathbb C[x_1,\ldots,x_n]_d=\mathbb C[V]_d = \operatorname{Sym}^d V^\ast$, the $d$-th symmetric power of $V^\ast$. I am setting up this notation so that the wiki page is more comprehensible.
Now $G$ acts on $\mathbb C[V]$ via the action $G\times\mathbb C[V]\to\mathbb C[V]$ defined by $g.\phi:=\phi\circ\rho(g)^{-1}$. Homogeneous polynomials of degree $d$ are mapped to homogeneous polynomials of degree $d$ under this action, so $\mathbb C[V]_d$ is a finite dimensional $G$-representation. If we denote by $n_d := \dim_{\mathbb C}(\mathbb C[V]_d^G)$ the dimension of its invariant subspace, then the Molien series is the generating function of the sequence $d\mapsto n_d$. Note that a generating function of a sequence is a general concept, it is the power series $H(T)=\sum_{d=0}^\infty n_d T^d$ in the variable $T$.
Geometrically, $\mathbb C[V]_d^G=\mathbb C[V/G]_d$ is the $d$-th graded part of the coordinate ring of the geometric quotient $V/G$, so $H(T)$ is the Hilbert series of the variety $V/G$.
Did this explain how everything is related? Feel free to ask.
