Polynomial invariant ring
= Polynomial invariant ring
{title2=$\mathbb C[W]^G$}
For a group acting linearly on $W$, act on the <coordinate ring> by $(g\cdot f)(w)=f(g^{-1}w)$. The <polynomial invariant ring> is the fixed subalgebra $\mathbb C[W]^G$. It inherits the degree grading. Even over the complex numbers it need not be a <unique factorization domain>, as the <quadratic cone invariant ring> demonstrates.