for all $n \geq 0$, and for all $n \gg 0$:

1.  $\cap_i \ker\phi^i_n = \{0\}$;

2.  $\sum_i \phi_n^i(W_n)$ spans $W_{n+1}$ as an ${\mathfrak{S}}_{n+1}$-representation;

3.  for any partition $\lambda$, and any integers $n_1 \geq \ldots \geq n_d \geq |\lambda| + \lambda_1$, let $c_{\lambda,n_1,\ldots,n_d}$ be the multiplicity of $S(\lambda)_{n_1,\ldots,n_d}$ in $W_{\sum_i n_i - (d-1)|\lambda|}$. Then the quantity $c_{\lambda,n_1+l,\ldots,n_d+l}$ is independent of $l$ for $l \gg 0$.\

Moreover, if $W$ is finitely generated, and $c_{\lambda,n}$ is the multiplicity of $S(\lambda)_n$ in $W_n$, then there exists a polynomial $p(x) \in {\mathbb{Q}}[x]$ of degree $\leq d-1$ such that for all $n \gg 0$, $c_{\lambda,n} = p(n)$. Finally, if $W$ is finitely generated then there exists an integer $b_W$, independent of $n$, such that if $S^\lambda$ is an irreducible representation appearing in $W_n$, then $\lambda$ has at most $b_W$ boxes below its $d$-th row.\

Having stated Theorems \[noeth\] and \[genrepstab\], one sees that Theorems \[sinkexpstab\] and \[sinkrepstab\] will follow from the modules $H_q(\operatorname{Conf}^{\operatorname{sink}}_n(G,V);{\mathbb{Q}})$ forming a finitely generated $\operatorname{FI}_V$-module. While we will need more machinery before we can prove finite generation, we can at least present this $\operatorname{FI}_V$-module structure now.\

\[fivaction\] Let $(f,g):[m] \rightarrow [n]$ be a morphism in $\operatorname{FI}_V$. Then we define an induced morphism 