Invariant polynomial
From HandWiki
In mathematics, an invariant polynomial is a polynomial [math]\displaystyle{ P }[/math] that is invariant under a group [math]\displaystyle{ \Gamma }[/math] acting on a vector space [math]\displaystyle{ V }[/math]. Therefore, [math]\displaystyle{ P }[/math] is a [math]\displaystyle{ \Gamma }[/math]-invariant polynomial if
- [math]\displaystyle{ P(\gamma x) = P(x) }[/math]
for all [math]\displaystyle{ \gamma \in \Gamma }[/math] and [math]\displaystyle{ x \in V }[/math].[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
 |  |
