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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?

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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.

Jesko Hüttenhain
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  • yes, I have few questions. The explanation is quite inaccessible to me; I felt third and fourth paragraphs are more technical at this stage for me. In simpler words, how does Molien series describe the polynomial invariants of the group? Does the Molien series say anything about the uniqueness of the invariant polynomial? For instance, if the finite group is perfect, can we say anything about the uniqueness (i.e. upto multiplication by scalars). In my case, my group is Valentiner group $G$, constructed as a perfect $3:1$ central extension of $A_6$. Here $G$ is perfect too. TBC... –  Dec 03 '14 at 23:01
  • C'td: Now I want to understand the polynomial invariants of $G$. In particular how does Molien series explain the invariant polynomial, its uniqueness perhaps... –  Dec 03 '14 at 23:03
  • @monomorphic: The Molien series does not really describe the polynomial invariants. I do not think it is possible to reconstruct the polynomial invariants from the Molien series. The Molien series is really just a power series whose $d$-th coefficient is the number of linearly independent polynomial invariants of degree $d$. What uniqueness are you referring to? – Jesko Hüttenhain Dec 03 '14 at 23:04
  • I understand something from your edited comment. Let me think more about this. Then I will be back. –  Dec 03 '14 at 23:12
  • @monomorphic: Alright =) – Jesko Hüttenhain Dec 03 '14 at 23:12
  • wow! So if the Molien series associated to a group $G$ is $$1+t^6+2t^{12}+2t^{18}+...$$ then we can immediately conclude that $G-$invariant polynomial of degree $6$ is unique upto multiplication by a scalar, number of linearly independent $G-$invariant polynomials of degree $12$ is $2$, and so on. Correct? This is remarkable! Could you provide some basic idea (just to have some feeling about why its true) about the proof of Molien's theorem? –  Dec 03 '14 at 23:27
  • @monomorphic: You got it. Unfortunately, I am afraid I have never seen a proof of Molien's theorem and you might best be served asking this as another question, I am sure someone here can help you with that. – Jesko Hüttenhain Dec 03 '14 at 23:38
  • No problem! This explanation of yours is already enough for me at this stage. Thank you so much! –  Dec 03 '14 at 23:45
  • Huttenhain : Suppose $V$ is an irreducible representation of the symmetric group $S_n$ say corresponding to the partion (n-1,1). How can we compute the Molien series for this explicitely? – budi Dec 07 '16 at 05:00