Finite linear invariant and coinvariant polynomial algebras

Let $V$ be a finite-dimensional complex vector space and $G\le \mathrm{GL}(V)$ finite. Write $S=\mathbb{C}[V]=\mathrm{Sym}(V^{\ast })$ for its polynomial algebra. Concretely, after a finite choice of linear coordinates $x_1,\dots ,x_r$, this is the iterated polynomial ring $\mathbb{C}[x_1,\dots ,x_r]$ of The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution, with finite monomial sums, total degree and substitution of linear forms. Under a change of coordinates, the two invertible linear substitutions are inverse algebra homomorphisms, so the construction is independent of that choice.

Define $(g\cdot f)(v)=f(g^{-1}v)$. Substitution respects sums and products, preserves total degree, and $(g\cdot (h\cdot f))(v)=f(h^{-1}{g}^{-1}v)=((gh)\cdot f)(v)$, so this is an action by graded algebra automorphisms. Let $R=S^G=\{f\in S:g\cdot f=f\text{ for every }g\in G\},R_{+}={⨁}_{d>0}{R}_d,I=S{R}_{+}\mathrm{.}$ The invariant polynomial algebra is $R$; the coinvariant algebra is the graded quotient $S\mathrm{/}I$. The notation $S{R}_{+}$ means finite sums of products $fp$ with $f\in S$ and $p\in R_{+}$. Because the action is graded, homogeneous parts of an invariant are invariant. Thus $R_{+}$ is an ideal of $R$, $I$ is a homogeneous ideal of $S$, and the quotient grading is well-defined.

The Reynolds operator is $\mathcal{R}(f)=\mathrm{\mid }G{\mathrm{\mid }}^{-1}{\sum }_{g\in G}g\cdot f$. The group is nonempty and $\mathrm{\mid }G\mathrm{\mid }$ is invertible in $\mathbb{C}$. Left multiplication permutes its finite elements, so every $\mathcal{R}(f)$ is invariant. For invariant $f$ all summands equal $f$, hence $\mathcal{R}(f)=f$ and ${\mathcal{R}}^2=\mathcal{R}$. For $p\in R$, $\mathcal{R}(pf)=p\mathcal{R}(f)$ termwise. It is therefore a graded $R$-linear projection onto $R$.

The polynomial-function notation is faithful over $\mathbb{C}$: a univariate nonzero degree-$d$ polynomial has at most $d$ roots by repeated division by $x-a$, and induction on the number of variables, viewing the last variable's coefficients as polynomials in the others, shows that a polynomial vanishing everywhere is zero. Thus the substitution formulas can be checked either formally or on points.

If $V=0$, then $S=R=\mathbb{C}$, $G$ is the trivial subgroup, $R_{+}=I=0$ and the coinvariant algebra is $\mathbb{C}$. If $G=\{1\}$ in positive dimension, $R=S$ and $I=(x_1,\dots ,x_r)$, so again $S\mathrm{/}I=\mathbb{C}$. Constants survive in every case because $I$ has positive degree. All averages and coordinate choices are finite; no AC is used.