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

${\displaystyle P(\gamma x)=P(x)}$

for all ${\displaystyle \gamma \in \Gamma }$ and ${\displaystyle x\in V}$.^[1]

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.^[2]