Schur Weyl duality in complex rank

Abstract

This thesis 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(St) equipped with a natural action of gl(V). This construction allows us to describe a duality between the abelian envelope of the category Rep(St) and a localization of the category Op/t,v (the parabolic category 0 for gl(V) associated with the pair (V, 1)). In particular, we obtain an exact contravariant functor SWt from the category Repab(St) (the abelian envelope of the category Rep(St)) to a certain quotient of the category Op/t v. This quotient, denoted by 0 p/t v, is obtained by taking the full subcategory of Op/t v consisting of modules of degree t, and localizing by the subcategory of finite dimensional modules. It turns out that the contravariant functor SWt makes Op/t v a Serre quotient of the category Repab(St)OP, and the kernel of SWt can be explicitly described. In the second part of this thesis, we consider the case when V = C[infinity] . We define the appropriate version of the parabolic category 0 and its localization, and show that the latter is equivalent to a "restricted" inverse limit of categories Op/t1CN with N tending to infinity. The Schur-Weyl functors SWt,CN then give an anti-equivalence between the category Op[infinity]/t C[infinity]and the category Repab(Se). This duality provides an unexpected tensor structure on the category Op[infinity]/t C[infinity].