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_∞) of algebraic representations of the infinite symmetric group S_∞=⋃_nS_n, related to the theory of FI-modules. The other is the family of rigid symmetric monoidal Deligne categories Rep_(S_t), t∈C, together with their abelian versions Rep_^ab(S_t), constructed by Comes and Ostrik. We show that for any t∈C the natural functor Rep(S_∞)→Rep_^ab(S_t) is an exact symmetric faithful monoidal functor, and compute its action on the simple representations of S_∞. Considering the highest weight structure on Rep_^ab(S_t), we show that the image of any object of Rep(S_∞) 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.