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 formula_1, 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: Here, formula_3 is the  subspace of formula_4 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. Derivation. Let formula_5 denote the irreducible characters of a finite group "G" and "V", "R" as above. Then the character formula_6 of formula_4 can be written as: Here, each formula_9 is given by the inner product: where formula_11 and formula_12 are the possibly repeated eigenvalues of formula_13. Now, we compute the series : Taking formula_15 to be the trivial character yields Molien's formula. Example. Consider the symmetric group formula_16 acting on R by permuting the coordinates. We add up the sum by group elements, as follows. Starting with the identity, we have There is a three-element conjugacy class of formula_16, consisting of swaps of two coordinates. This gives three terms of the form There is a two-element conjugacy class of cyclic permutations, yielding two terms of the form Notice that different elements of the same conjugacy class yield the same determinant. Thus, the Molien series is On the other hand , we can expand the geometric series and multiply out to get The coefficients of the series tell us the number of linearly independent homogeneous polynomials in three variables which are invariant under permutations of the three variables, i.e. the number of independent symmetric polynomials in three variables. In fact, if we consider the elementary symmetric polynomials we can see for example that in degree 5 there is a basis consisting of formula_26, formula_27, formula_28, formula_29, and formula_30.