Equivalences between blocks of p-local Mackey algebras

TL;DR

This paper investigates the relationships between blocks of p-local Mackey algebras and group algebras, proving Broué-type equivalences in specific cases like p-nilpotent groups and defect 1 blocks.

Contribution

It establishes that Broué's conjecture analogues hold for p-local Mackey algebras in certain cases, expanding the understanding of block equivalences in modular representation theory.

Findings

Proves Broué-type equivalences for principal blocks of p-nilpotent groups.

Establishes equivalences for blocks with defect 1.

Highlights the potential role of splendid equivalences in Mackey algebras.

Abstract

Let $G$ be a finite group and $(K,\mathcal{O},k)$ be a $p$-modular system. Let $R=\mathcal{O}$ or $k$. There is a bijection between the blocks of the group algebra and the blocks of the so-called $p$-local Mackey algebra $\mu_{R}^{1}(G)$. Let $b$ be a block of $RG$ with abelian defect group $D$. Let $b'$ be its Brauer correspondant in $N_{G}(D)$. It is conjectured by Brou\'e that the blocks $RGb$ and $RN_{G}(D)b'$ are derived equivalent. Here we look at equivalences between the corresponding blocks of $p$-local Mackey algebras. We prove that an analogue of the Brou\'e's conjecture is true for the $p$-local Mackey algebras in the following cases: for the principal blocks of $p$-nilpotent groups and for blocks with defect $1$. We also point out the probable importance of \emph{splendid} equivalences for the Mackey algebras.