Chevalley shephard todd for finite weyl groups

Apply F1 to the given finite subgroup: each generating reflection is precisely a complex reflection under the stated hyperplane definition. If its minimal invariant ideal generators are $f_1,\dots ,f_m$, the map $\mathbb{C}[t_1,\dots ,t_m]\to S^W$, $t_i\mapsto f_i$, is surjective by algebra generation and injective by algebraic independence. Giving $t_i$ degree $\mathrm{deg}{f}_i$ makes this a graded algebra isomorphism.

Under the stated AC, F2 gives $m=r$ and a finite homogeneous basis of $S\mathrm{/}(f_1,\dots ,f_r)S$ whose arbitrary homogeneous lifts form a free $S^W$-basis. F3 says this basis has $\mathrm{\mid }W\mathrm{\mid }$ elements, proving the asserted rank as well as polynomiality. These suppliers concern the same ideal $S{S}_{+}^W$ and the same faithful action; no assertion about a Lie algebra restriction map is involved.

When $r=0$, the subgroup of $\mathrm{GL}(0)$ is trivial, the invariant algebra is the polynomial algebra in zero variables $\mathbb{C}$, and its module rank is one. Fixed directions and degree-one generators are allowed by F1 and F2. Only a finite coinvariant basis and finitely many lifts are selected here; the stated AC premise is needed for the current cited proofs of F2 and F3. This proves the stated reflection-group direction; no converse characterizing arbitrary invariant-polynomial groups is asserted.