invariant polynomial

An invariant polynomial is a polynomial $P$ that is invariant under a (compact) Lie group $\mathrm{\Gamma }$ acting on a vector space $V$. Therefore $P$ is $\mathrm{\Gamma }$-invariant polynomial if $P(\gamma x)=P(x)$ for all $\gamma \in \mathrm{\Gamma }$ and $x\in V$.