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.[1]

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.[2]

References

  1. ^ "invariant polynomial in nLab". ncatlab.org
  2. ^ Draisma, Jan & Gijswijt, Dion. "Invariant Theory with Applications"