  We construct a complex $\mathcal{L}_\bullet^\lambda$ resolving the
irreducible representations $\mathcal{S}^{\lambda[n]}$ of the symmetric groups
$S_n$ by representations restricted from $GL_n(k)$. This construction lifts to
$\mathrm{Rep}(S_\infty)$, where it yields injective resolutions of simple
objects. It categorifies stable Specht polynomials, and allows us to understand
evaluations of these polynomials for all $n$.
