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Diagonal $p$-permutation functors, semisimplicity, and functorial equivalence of blocks

Let $k$ be an algebraically closed field of characteristic $p>0$, let $R$ be a commutative ring, and let $\mathbb{F}$ be an algebraically closed field of characteristic 0. We consider the $R$-linear category $\mathcal{F}^Δ_{Rpp_k}$ of diagonal $p$-permutation functors over $R$. We first show that the category $\mathcal{F}^Δ_{\mathbb{F}pp_k}$ is semisimple, and we give a parametrization of its simple objects, together with a description of their evaluations. Next, to any pair $(G,b)$ of a finite group $G$ and a block idempotent $b$ of $kG$, we associate a diagonal $p$-permutation functor $RT^Δ_{G,b}$ in $\mathcal{F}^Δ_{Rpp_k}$. We find the decomposition of the functor $\mathbb{F}T^Δ_{G,b}$ as a direct sum of simple functors in $\mathcal{F}^Δ_{\mathbb{F}pp_k}$. This leads to a characterization of nilpotent blocks in terms of their associated functors in $\mathcal{F}^Δ_{\mathbb{F}pp_k}$. Finally, for such pairs $(G,b)$ of a finite group and a block idempotent, we introduce the notion of functorial equivalence over $R$, which (in the case $R=\mathbb{Z}$) is slightly weaker than $p$-permutation equivalence, and we prove a corresponding finiteness theorem: for a given finite $p$-group $D$, there is only a finite number of pairs $(G,b)$, where $G$ is a finite group and $b$ a block idempotent of $kG$ with defect isomorphic to $D$, up to functorial equivalence over $\mathbb{F}$.