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

arXiv commentv3: minor corrections. To appear in Advances in Mathematics; v2: Corrected a mistake in section 6