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In [[mathematics]], an '''invariant polynomial''' is a [[polynomial]] <math>P</math> that is [[invariant (mathematics)|invariant]] under a [[group (mathematics)|group]] <math>\Gamma</math> [[group action|acting]] on a [[vector space]] <math>V</math>. Therefore <math>P</math> is a <math>\Gamma</math>-invariant polynomial if
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:<math>P(\gamma x) = P(x)</math>
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for all <math>\gamma \in \Gamma</math> and <math>x \in V</math>.
 
Cases of particular importance are for &Gamma; 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 power]]s of the given [[linear representation]] of &Gamma;.
 
==References==
* {{PlanetMath attribution|id=4337|title=Invariant polynomial}}
 
[[Category:Commutative algebra]]
[[Category:Invariant theory]]
[[Category:Polynomials]]
{{algebra-stub}}

Latest revision as of 18:35, 14 December 2014

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