Simple commutative algebras in Deligne's categories Rep($S_t$)

Abstract

We show that in the Deligne categories $\mathrm{Rep}(S_t)$ for $t$ a transcendental number, the only simple algebra objects are images of simple algebras in the category of representations of a symmetric group under a canonical induction functor. They come in families which interpolate the families of algebras of functions on the cosets of $H\times S_{n-k}$ in $S_n$, for a fixed subgroup $H$ of $S_k$.

(Changelog: Remove theorem 5.6)

Cited by in corpus (5)

• Deligne categories as limits in rank and characteristic
• Semisimple and $G$-equivariant simple algebras over operads
• Harish-Chandra bimodules in the Deligne category $\text{Rep}(G{L}_t)$
• Classification of simple algebras in Deligne category $Rep(S_t)$
• Harish-Chandra bimodules of finite $K$-type in Deligne categories