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].

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 [[PlanetMath:4337|Invariant polynomial]] on PlanetMath, which is licensed under the [Text of the GNU Free Documentation LicenseGFDL].

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