Molien series

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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.

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 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 subspaces fixed by all elements of G, and n_d is its dimension.

The Molien series is then by definition the formal power series

Σ n_dt^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.