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In mathematics, an invariant polynomial is a polynomial P {\displaystyle P} that is invariant under a group Γ {\displaystyle \Gamma } acting on a vector space V {\displaystyle V}. Therefore, P {\displaystyle P} is a Γ {\displaystyle \Gamma } -invariant polynomial if
for all γ ∈ Γ {\displaystyle \gamma \in \Gamma } and x ∈ V {\displaystyle x\in V}.
Cases of particular importance are for Γ a finite group , 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.
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