Invariant polynomial

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

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

for all $\gamma \in \mathrm{\Gamma }$ 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

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