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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 R n = C n = Sym n {\displaystyle R_{n}=\mathbb {C} _{n}=\operatorname {Sym} ^{n}} , the space of homogeneous polynomial functions on V of degree n , if G is a finite group, the series can be computed as:
Here, R n G {\displaystyle R_{n}^{G}} is the subspace of R n {\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.