Stable functorial equivalence of blocks

Abstract

Let $k$ be an algebraically closed field of characteristic $p>0$, let $R$ be a commutative ring and let $\mathcal{F}$ be an algebraically closed field of characteristic $0$. We introduce the category $\overline{{\mathcal{F}}_{Rp{p}_k}}$ of stable diagonal $p$-permutation functors over $R$. We prove that the category $\overline{{\mathcal{F}}_{\mathbb{F}p{p}_k}}$ is semisimple and give a parametrization of its simple objects in terms of the simple diagonal $p$-permutation functors. We also introduce the notion of a stable functorial equivalence over $R$ between blocks of finite groups. We prove that if $G$ is a finite group and if $b$ is a block idempotent of $kG$ with an abelian defect group $D$ and Frobenius inertial quotient $E$, then there exists a stable functorial equivalence over $\mathbb{F}$ between the pairs $(G,b)$ and $(D⋊E,1)$.

arXiv admin note: text overlap with arXiv:2212.02054