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

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

for all ${\displaystyle \gamma \in \Gamma }$ and $x\in V$.

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.

References

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