# Invariant polynomial

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In [mathematics](/source/Mathematics), an **invariant polynomial** is a [polynomial](/source/Polynomial) P that is [invariant](/source/Invariant_(mathematics)) under a [group](/source/Group_(mathematics)) \Gamma [acting](/source/Group_action_(mathematics)) on a [vector space](/source/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](/source/Finite_group) (in the theory of [Molien series](/source/Molien_series), in particular), a [compact group](/source/Compact_group), a [Lie group](/source/Lie_group) or [algebraic group](/source/Algebraic_group). For a basis-independent definition of 'polynomial' nothing is lost by referring to the [symmetric powers](/source/Symmetric_power) of the given [linear representation](/source/Linear_representation) of Γ.[2]

## References

1. ["invariant polynomial in nLab"](https://ncatlab.org/nlab/show/invariant+polynomial). *ncatlab.org*

1. Draisma, Jan & Gijswijt, Dion. ["Invariant Theory with Applications"](http://www.win.tue.nl/~jdraisma/teaching/invtheory0910/lecturenotes11.pdf)

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Adapted from the Wikipedia article [Invariant polynomial](https://en.wikipedia.org/wiki/Invariant_polynomial) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Invariant_polynomial?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.