The basic degrees are independent of the chosen family; Hilbert series of the invariants and of the coinvariant algebra; the order formula and the Molien identity

Put $\mathfrak{m}=S_{+}=(x_1,\dots ,x_n)$. For any finite minimal homogeneous generating family $g_1,\dots ,g_r$ of $I$, its classes span the graded complex vector space $I\mathrm{/}\mathfrak{m}I$: modulo $\mathfrak{m}I$, coefficients in an expression ${\sum }_j{a}_j{g}_j$ reduce to their constants. These classes are independent. Indeed, a nonzero homogeneous relation would give ${\sum }_{\mathrm{deg}{g}_j=D}{c}_j{g}_j\in \mathfrak{m}I$ with some $c_k\mathrm{\ne }0$. Since the $g_j$ generate $I$, its right side can be written ${\sum }_j{b}_j{g}_j$ with $b_j\in \mathfrak{m}$ homogeneous of degree $D-\mathrm{deg}{g}_j$. Such $b_j$ vanish when $\mathrm{deg}{g}_j\ge D$, so solving for $g_k$ expresses it in the ideal generated by the other $g_j$, contrary to minimality. Every relation splits into homogeneous ones, proving independence. Thus the number of degree-$D$ members in any such family is ${\dim }_{\mathbb{C}}(I\mathrm{/}\mathfrak{m}I{)}_D$. Applying this to the fixed invariant family and to an arbitrary other minimal homogeneous family gives equality of degree multisets and family sizes. This proves (1) without assuming that the other generators are invariant; for $I=0$ the minimal family and quotient are empty.

For the fixed invariant basic family, [F1] gives the graded isomorphism $\mathbb{C}[y_1,\dots ,y_n]\to R$ with $y_i\mapsto f_i$ and $\mathrm{deg}{y}_i=d_i$. Counting its monomials gives $\mathrm{Hilb}(R,t)={\sum }_{N\ge 0}\mathrm{\#}\{a\in {\mathbb{N}}^n:{\sum }_i{a}_i{d}_i=N\}{t}^N={\prod }_{i=1}^n(1-t^{d_i}{)}^{-1}$. Positive weights make every coefficient finite, so the product is a formal-power-series identity. This algebra isomorphism is asserted only for the invariant basic family.

Since $I$ is homogeneous, $R=⨁R_N$ and $A=⨁A_N$ are graded with finite-dimensional pieces, so the Hilbert series of [F5] apply. By step 1.2, $\mathrm{Hilb}(R,t)={\prod }_{i=1}^n(1-t^{d_i}{)}^{-1}$. Under the stated AC the group ${\rho }_{\mathbb{C}}(W)$ is a finite complex reflection group of order $\mathrm{\mid }W\mathrm{\mid }$ on the $n$-dimensional space $V_{\mathbb{C}}$ with invariant algebra $R$ and positive-degree ideal $I$, so [F3] gives $\mathrm{Hilb}(A,t)={\prod }_{i=1}^n(1+t+\cdots +t^{d_i-1})$, and evaluating at $t=1$ gives ${\dim }_{\mathbb{C}}A={\prod }_i{d}_i=\mathrm{\mid }W\mathrm{\mid }$; since each factor $1+t+\cdots +t^{d_i-1}$ is a polynomial of degree $d_i-1$, the product is a polynomial of degree ${\sum }_i{e}_i$ with leading coefficient $1$, so ${\sum }_i{e}_i$ is the top degree of $A$.

Fix $w\in W$. By [F2] and [F6] the operator induced by $w$ on the finite-dimensional space $V_{\mathbb{C}}^{\ast }$ is diagonalisable; let ${\lambda }_1,\dots ,{\lambda }_n$ be its eigenvalues with multiplicity, and note that $\det (1-tw\mid V_{\mathbb{C}}^{\ast })={\prod }_{j=1}^n(1-{\lambda }_{j}t)$. The induced operator on $S_N={\mathrm{Sym}}^N(V_{\mathbb{C}}^{\ast })$ ($N\ge 0$) has, in the monomial basis attached to an eigenbasis of $V_{\mathbb{C}}^{\ast }$, the eigenvalues ${\lambda }^a={\lambda }_1^{a_1}\cdots {\lambda }_n^{a_n}$ over all $a$ with $\mathrm{\mid }a\mathrm{\mid }=N$; summing gives the formal identity ${\sum }_{N\ge 0}\mathrm{tr}(w\mid S_N)t^N={\prod }_{j=1}^n(1-{\lambda }_{j}t{)}^{-1}=\det (1-tw\mid V_{\mathbb{C}}^{\ast }{)}^{-1}$ [F5]. The Reynolds operator is a projection of $S_N$ onto $R_N$ [F4], and for a projection the trace equals the dimension of its image, so ${\dim }_{\mathbb{C}}{R}_N=\mathrm{\mid }W{\mathrm{\mid }}^{-1}{\sum }_{w\in W}\mathrm{tr}(w\mid S_N)$; summing over $N$ and using the previous identity gives $\mathrm{Hilb}(R,t)=\mathrm{\mid }W{\mathrm{\mid }}^{-1}{\sum }_{w\in W}\det (1-tw\mid V_{\mathbb{C}}^{\ast }{)}^{-1}$, and combining with $\mathrm{Hilb}(R,t)={\prod }_i(1-t^{d_i}{)}^{-1}$ of 2.1 proves the Molien identity (4).

For $n=0$ the group is trivial, $S=R=\mathbb{C}$, $I=0$, $A=\mathbb{C}$, and every displayed product is empty and equals $1$, so all four clauses hold in this empty form [F1, F5]. Let now $\mathrm{\Gamma }$ have finitely many connected components and suppose each is of finite type; by [F7] $W\cong W_1\times \cdots \times W_k$ and $V_{\mathbb{C}}=V_{1,\mathbb{C}}\oplus \cdots \oplus V_{k,\mathbb{C}}$ with $\rho (W_i)$ preserving $V_i$ and fixing the other summands. Choose coordinates adapted to this decomposition, so that $S=\mathbb{C}[x^{(1)},\dots ,x^{(k)}]$ with $x^{(i)}$ the coordinates of the $i$-th summand [F8]. For $f\in R=S^W$ and any $i$, the Reynolds operator ${\mathcal{R}}_i$ of $W_i$ acts only on the block $x^{(i)}$ and, because $f$ is $W_i$-invariant, leaves it unchanged; expanding $f$ in the monomials of the remaining blocks and applying ${\mathcal{R}}_i$ expresses $f$ as a finite sum of products of a $W_i$-invariant polynomial in the block $x^{(i)}$ with a polynomial in the other blocks. Applying ${\mathcal{R}}_k,{\mathcal{R}}_{k-1},\dots ,{\mathcal{R}}_1$ successively (each application leaves $f$ unchanged and replaces one block factor by its invariant part) gives $f\in \mathbb{C}[x^{(1)}{]}^{W_1}\cdots \mathbb{C}[x^{(k)}{]}^{W_k}$: the invariant algebra is generated by the component invariant algebras [F4, F8]. If for each $i$ we fix a basic family of the component $W_i$, then the concatenated family is homogeneous of positive degrees, generates $R$, and is algebraically independent: generation follows from the preceding display, and a polynomial relation among the concatenated family, expanded in the block monomials and using the algebraic independence of each component's family, forces every coefficient polynomial to vanish [F1, F8]. The concatenation generates $I=S{R}_{+}$ because its members generate $R$ and have positive degrees. It is also minimal as an $S$-ideal generating family: any redundancy, after Reynolds averaging its coefficients [F4], would express one member as an $R$-linear combination of the others. Substituting their polynomial expressions in the algebraically independent family and setting all the other variables to zero would give the impossible identity $y_j=0$ in $\mathbb{C}[y_j]$. Thus the concatenation is a basic family of the reducible system, and its degree multiset is the concatenation of the components' multisets; by the multiset independence of 1.1 this is the degree multiset of every basic family of the reducible system. All statements above are under the stated AC, which enters only through the existence of the $n$-element basic families and the AC-scoped suppliers [F1, F3]; the series comparison, the trace computation and the componentwise argument are finite and choice-free.