Littlewood complexes for symmetric groups

Abstract

We construct a complex L ∙ λ \mathcal {L}_\bullet ^\lambda resolving the irreducible representations S λ [ n ] \mathcal {S}^{\lambda [n]} of the symmetric groups S n S_n by representations restricted from G L n ( k ) GL_n(k) . This construction lifts to R e p ( S ∞ ) \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 n .