# Molien series

From Groupprops

*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 be a group and be a finite-dimensional linear representation of . The **Molien series** of is the formal power series:

where is the dimension of the subspace of comprising those elements that are invariant under the action of .

Equivalently, the map gives an action of on the polynomial ring in variables. The dimension of the space of homogeneous degree polynomials that are invariant under this action, is the coefficient of .

## Facts

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