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- W4302360147 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 $n$." @default.
- W4302360147 created "2022-10-06" @default.
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- W4302360147 date "2017-06-09" @default.
- W4302360147 modified "2023-09-27" @default.
- W4302360147 title "Deligne categories and representations of the infinite symmetric group" @default.
- W4302360147 doi "https://doi.org/10.48550/arxiv.1706.03645" @default.
- W4302360147 hasPublicationYear "2017" @default.
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