Molien series

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This term makes sense in the context of a linear representation of a group, viz an action of the group as linear automorphisms of a vector space


Let G be a group and \rho:G \to GL(V) be a finite-dimensional linear representation of G. The Molien series of \rho is the formal power series:

\sum n_d t^d

where n_d is the dimension of the subspace of Sym^d(V) comprising those elements that are invariant under the action of G.

Equivalently, the map \rho:G \to GL(V) gives an action of G on the polynomial ring in dim(V) variables. The dimension of the space of homogeneous degree d polynomials that are invariant under this action, is the coefficient n_d of t^d.


For a finite group, the Molien series corresponding to any finite-dimensional representation is a rational function of the formal variable t.

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