We establish a connection between two settings of representation stability for the symmetric groups Sn over C. One is the symmetric monoidal category Rep(S∞) of algebraic representations of the infinite symmetric group S∞=⋃nSn, related to the theory of FI-modules. The other is the family of rigid symmetric monoidal Deligne categories Rep_(St), t∈C, together with their abelian versions Rep_ab(St), constructed by Comes and Ostrik. We show that for any t∈C the natural functor Rep(S∞)→Rep_ab(St) 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(St), we show that the image of any object of Rep(S∞) has a filtration with standard objects in Rep_ab(St). 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_(St), and their specializations at non-negative integers n.
- Deligne categories
- Representations of the infinite symmetric group
- Stable representation theory
- Symmetric group
- Tensor categories