# Equivalences between blocks of p-local Mackey algebras.

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* Corresponding author 1 Théorie des groupes. LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée

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 Broué that the blocks $RGb$ and $RN {G}Db-$ 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.

Keywords : block theory Finite group Mackey functor block theory.

Autor: Baptiste Rognerud -

Fuente: https://hal.archives-ouvertes.fr/

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