Deligne categories and representations of the infinite symmetric group

We establish a connection between two settings of representation stability for the symmetric groups S-n over C. One is the symmetric monoidal category Rep(S-infinity) of algebraic representations of the infinite symmetric group S-infinity = U-n S-n, related to the theory of FI-modules. The other is the family of rigid symmetric monoidal Deligne categories Rep(S-t), t is an element of C, together with their abelian versions Rep(ab)(S-t), constructed by Comes and Ostrik. We show that for any t is an element of C the natural functor Rep(S-infinity) -> Rep(ab)(S-t) is an exact symmetric faithful monoidal functor, and compute its action on the simple representations of S-infinity. Considering the highest weight structure on Rep(ab)(S-t), we show that the image of any object of Rep(S-infinity) has a filtration with standard objects in Rep(ab)(S-t). As a by-product of the proof, we give answers to the questions posed by P. Deligne concerning the cohomology of some complexes in the Deligne category Rep(S-t), and their specializations at non-negative integers n. (C) 2019 Elsevier Inc. All rights reserved.