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

Formulation

More formally, there is a vector space of such polynomials, for each given value of d = 0, 1, 2, ..., and we write n_d for its vector space dimension, or in other words the number of linearly independent homogeneous invariants of a given degree. In more algebraic terms, take the d-th symmetric power of V, and the representation of G on it arising from ρ. The invariants form the subspace consisting of all vectors fixed by all elements of G, and n_d is its dimension.

The Molien series is then by definition the formal power series

$M(t)=\sum _d{n}_d{t}^d.$

This can be looked at another way, by considering the representation of G on the symmetric algebra of V, and then the whole subalgebra R of G-invariants. Then n_d is the dimension of the homogeneous part of R of dimension d, when we look at it as graded ring. In this way a Molien series is also a kind of Hilbert function. Without further hypotheses not a great deal can be said, but assuming some conditions of finiteness it is then possible to show that the Molien series is a rational function. The case of finite groups is most often studied.

Formula

Molien showed that

$M(t)=\frac{1}{\left|G\right|}\sum _{g\in G}\frac{1}{\det (I-tg)}$

This means that the coefficient of t^d in this series is the dimension n_d defined above. It assumes that the characteristic of the field does not divide |G| (but even without this assumption, Molien's formula in the form $\left|G\right|\cdot M\left(t\right)=\sum _{g\in G}\frac{1}{\det (I-tg)}$ is valid, although it does not help with computing M(t)).

Example

Consider $S_3$ acting on R³ by permuting the coordinates. Note that $\det (I-tg)$ is constant on conjugacy classes, so it is enough to take one from each of the three classes in $S_3$; so $\det (I-te)=(1-t{)}^3,\det (I-t{\sigma }_2)=(1-t)(1-t^2)$ and $\det (1-t{\sigma }_3)=(1-t^3)$ where ${\sigma }_2=(1,2)$ and ${\sigma }_3=(1,2,3)$.

Then

$M(t)=\frac{1}{6}(\frac{1}{(1-t{)}^3}+\frac{3}{(1-t)(1-t^2)}+\frac{2}{1-t^3})=\frac{1}{(1-t)(1-t^2)(1-t^3)}$

References

• David A. Cox, John B. Little, Donal O'Shea (2005), Using Algebraic Geometry, pp. 295–8
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