All simple commutative algebras in the Delannoy category are precisely those corresponding to certain transitive G-sets.
Simple commutative algebras in Deligne's categories Rep($S_t$)
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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 $S_k$.
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Classification of simple commutative algebras in the Delannoy category
All simple commutative algebras in the Delannoy category are precisely those corresponding to certain transitive G-sets.