Invariant polynomial

In mathematics, an invariant polynomial is a polynomial P that is invariant under a group Γ acting on a vector space V . Therefore P is a Γ -invariant polynomial if

for all γ ∈ Γ and x ∈ V .

Cases of particular importance are for Γ a finite group (in the theory of Molien series, in particular), a compact group, a Lie group or algebraic group. For a basis-independent definition of 'polynomial' nothing is lost by referring to the symmetric powers of the given linear representation of Γ.