Stable functorial equivalence of blocks

Serge Bouc and Deniz Yılmaz

Vol. 328 (2024), No. 2, 217–226

DOI: 10.2140/pjm.2024.328.217

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 introduce the category $\overline{{\mathcal{ℱ}}_{Rp{p}_k}^{\mathrm{\Delta }}}$ of stable diagonal $p$-permutation functors over $R$. We prove that the category $\overline{{\mathcal{ℱ}}_{\mathbb{𝔽}p{p}_k}^{\mathrm{\Delta }}}$ 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 $(D⋊E,1)$.

Keywords

block, diagonal $p$-permutation functors, functorial equivalence, Frobenius inertial quotient

Mathematical Subject Classification

Primary: 16S34, 20C20, 20J15

Milestones

Received: 5 November 2023

Accepted: 30 March 2024

Published: 30 April 2024