Invariant Polynomial

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

for all and .

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.