Littlewood complexes for symmetric groups

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 $n$.