Equivalences between blocks of p-local 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 $b^{\prime }$ be its Brauer correspondant in $N_G(D)$. It is conjectured by Broué that the blocks $RGb$ and $R{N}_G(D)b^{\prime }$ are derived equivalent. Here we look at equivalences between the corresponding blocks of $p$-local Mackey algebras. We prove that an analogue of the Broué'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.

arXiv comment24 pages. Second version. All the part about cohomological Mackey algebra has been remove