$k$ is a field of characteristic 0 as [@Sn Theorem 3.1]. The specific case of $\operatorname{FI}$-modules was also handled by Church, Ellenberg, Farb, and Nagpal in [@CEF] and [@CEFN].\
Specializing to the case where $k$ is a field of characteristic 0, we see that finitely generated $\operatorname{FI}_V$-modules can be examined from the perspective of the complex representation theory of the symmetric groups. Before we state our results, we first recall some notation.\

\[padpar\] Recall that the irreducible representations of the symmetric group ${\mathfrak{S}}_n$ over a field of characteristic 0 are in bijection with partitions $\lambda = (\lambda_1,\ldots,\lambda_h)$ of $n$. We will write $S^\lambda$ to denote this irreducible representation. If $\lambda$ is a partition of $n$ and $n_r \geq n_{d-1} \geq \ldots \geq n_1 \geq \lambda_1 + n$ is a sequence of integers, we define the **$r$-padded partition** $\lambda[n_1,\ldots,n_r] := (n_1-n,\ldots,n_r-n,\lambda_1,\ldots,\lambda_h)$. We will use $S(\lambda)_{n_1,\ldots,n_r} := S^{\lambda[n_1,\ldots,n_r]}$.\

\[genrepstab\] Assume that $|V| = d$. Let W be an $\operatorname{FI}_V$-module over a field $k$ of characteristic 0, and write $\phi^i_n:W_n \rightarrow W_{n+1}$ for the map induced by the pair of the standard inclusion $[n] \hookrightarrow [n+1]$ and the color $i$. Then $W$ is finitely generated if and only if $W_n$ is finite dimensional 