Deligne categories and representations of the infinite symmetric group

Abstract

We establish a connection between two settings of representation stability for the symmetric groups $S_n$ over $\mathbb{C}$. One is the symmetric monoidal category ${\rm Rep}(S_{\infty})$ of algebraic representations of the infinite symmetric group $S_{\infty} = \bigcup_n S_n$, related to the theory of ${\bf FI}$-modules. The other is the family of rigid symmetric monoidal Deligne categories $\underline{{\rm Rep}}(S_t)$, $t \in \mathbb{C}$, together with their abelian versions $\underline{{\rm Rep}}^{ab}(S_t)$, constructed by Comes and Ostrik. We show that for any $t \in \mathbb{C}$ the natural functor ${\rm Rep}(S_{\infty}) \to \underline{{\rm Rep}}^{ab}(S_t)$ is an exact symmetric faithful monoidal functor, and compute its action on the simple representations of $S_{\infty}$. Considering the highest weight structure on $\underline{{\rm Rep}}^{ab}(S_t)$, we show that the image of any object of ${\rm Rep}(S_{\infty})$ has a filtration with standard objects in $\underline{{\rm Rep}}^{ab}(S_t)$. 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 $\underline{{\rm Rep}}(S_t)$, and their specializations at non-negative integers $n$.