# Molien series

This term makes sense in the context of a linear representation of a group, viz an action of the group as linear automorphisms of a vector space

## Definition

Let $G$ be a group and $\rho:G \to GL(V)$ be a finite-dimensional linear representation of $G$. The Molien series of $\rho$ is the formal power series:

$\sum n_d t^d$

where $n_d$ is the dimension of the subspace of $Sym^d(V)$ comprising those elements that are invariant under the action of $G$.

Equivalently, the map $\rho:G \to GL(V)$ gives an action of $G$ on the polynomial ring in $dim(V)$ variables. The dimension of the space of homogeneous degree $d$ polynomials that are invariant under this action, is the coefficient $n_d$ of $t^d$.

## Facts

For a finite group, the Molien series corresponding to any finite-dimensional representation is a rational function of the formal variable $t$.