invariant polynomial

In mathematics, an invariant polynomial is a polynomial $P$ that is invariant under a group $\backslash Gamma$ acting on a vector space $V$. Therefore, $P$ is a $\backslash Gamma$-invariant polynomial if

:$P(\backslash gamma\; x)\; =\; P(x)$

for all $\backslash gamma\; \backslash in\; \backslash Gamma$ and $x\; \backslash in\; V$.{{cite web|title=invariant polynomial in nLab|url=https://ncatlab.org/nlab/show/invariant+polynomial|website=ncatlab.org}}

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.{{cite web|last1=Draisma|first1=Jan|last2=Gijswijt|first2=Dion|title=Invariant Theory with Applications|url=http://www.win.tue.nl/~jdraisma/teaching/invtheory0910/lecturenotes11.pdf}}