Invariant polynomial

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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

P(\gamma x)=P(x)

for all $\gamma \in \Gamma$ and $x\in V$ .^[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]

References

^ "invariant polynomial in nLab". ncatlab.org.
^ Draisma, Jan; Gijswijt, Dion. "Invariant Theory with Applications" (PDF).

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

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[1] "invariant polynomial in nLab". ncatlab.org.

[2] Draisma, Jan; Gijswijt, Dion. "Invariant Theory with Applications" (PDF).

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