Littlewood Complexes for Symmetric Groups

Abstract

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 $G{L}_n(k)$. 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 $n$.

9 pages