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 $n$.