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

Abstract

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 ${\mathcal{F}}_{Rp{p}_k}^{\Delta }$ of diagonal $p$-permutation functors over $R$. We first show that the category ${\mathcal{F}}_{\mathbb{F}p{p}_k}^{\Delta }$ 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 $R{T}_{G,b}^{\Delta }$ in ${\mathcal{F}}_{Rp{p}_k}^{\Delta }$. We find the decomposition of the functor $\mathbb{F}{T}_{G,b}^{\Delta }$ as a direct sum of simple functors in ${\mathcal{F}}_{\mathbb{F}p{p}_k}^{\Delta }$. This leads to a characterization of nilpotent blocks in terms of their associated functors in ${\mathcal{F}}_{\mathbb{F}p{p}_k}^{\Delta }$. 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 $\mathbb{F}$.