Invariant polynomial

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 ${\displaystyle x\in V}$.

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.

This article incorporates material from Invariant polynomial on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.