Hellenica World

# .

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.

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