Invariant polynomial

In mathematics, an invariant polynomial is a polynomial formula_1 that is invariant under a group formula_2 acting on a vector space formula_3. Therefore, formula_1 is a formula_2-invariant polynomial if
for all formula_7 and formula_8.
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.