Molien's formula

In mathematics, Molien's formula computes the generating function attached to a linear representation of a group G on a finite-dimensional vector space, that counts the homogeneous polynomials of a given total degree that are invariants for G. It is named for Theodor Molien.

Precisely, it says: given a finite-dimensional complex representation V of G and ${\displaystyle R_{n}=\mathbb {C} [V]_{n}=\operatorname {Sym} ^{n}(V^{*})}$, the space of homogeneous polynomial functions on V of degree n (degree-one homogeneous polynomials are precisely linear functionals), if G is a finite group, the series (called Molien series) can be computed as:^[1]

${\displaystyle \sum _{n=0}^{\infty }\dim(R_{n}^{G})t^{n}=(\#G)^{-1}\sum _{g\in G}\det(1-tg|V^{*})^{-1}.}$

Here, ${\displaystyle R_{n}^{G}}$ is the subspace of ${\displaystyle R_{n}}$ that consists of all vectors fixed by all elements of G; i.e., invariant forms of degree n. Thus, the dimension of it is the number of invariants of degree n. If G is a compact group, the similar formula holds in terms of Haar measure.