Invariant polynomial

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


 * $$P(\gamma x) = P(x)$$

for all $$\gamma \in \Gamma$$ and $$x \in V$$.

Cases of particular importance are for &Gamma; 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 &Gamma;.