Invariant polynomial

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

P ( γ x ) = P ( x ) {\displaystyle P(\gamma x)=P(x)}

for all γ ∈ Γ {\displaystyle \gamma \in \Gamma } and x ∈ V {\displaystyle 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" (PDF).

This article incorporates material from Invariant polynomial on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.


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