Schur-Weyl duality for Deligne categories

Abstract

This paper gives an analogue to the classical Schur-Weyl duality in the setting of Deligne categories. Given a finite-dimensional unital vector space V (i.e. a vector space V with a distinguished non-zero vector 1), we give a definition of a complex tensor power of V. This is an Ind-object of the Deligne category Rep(S_t), equipped with a natural action of gl(V). This construction allows us to describe a duality between the abelian envelope of the category Rep(S_t) and a localization of the parabolic category O for gl(V) associated with the pair (V, 1). In particular, we obtain an exact contravariant functor SW from the category Rep^{ab}(S_t) (the abelian envelope of the category Rep(S_t)) to a certain quotient \hat{O} of the parabolic category O. This quotient is obtained by taking the full subcategory consisting of modules of degree t, and localizing by the subcategory of finite dimensional modules. It turns out that the contravariant functor SW makes \hat{O} a Serre quotient of the category Rep^{ab}(S_t)^{op}, and the kernel of SW can be explicitly described.