Equivalences between blocks of p-local Mackey algebras.

Baptiste Rognerud (1)

1 Théorie des groupes.

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 $RN_{G}(D)b'$ are derived equivalent. Here we look at equivalences between the corresponding blocks of $p$-local Mackey algebras. We prove that an analogue of the 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.

Mots clés: Finite group; Mackey functor; block theory.; block theory

Domaines: Théorie des représentations [math.RT]; Théorie des groupes [math.GR]

Baptiste Rognerud. Equivalences between blocks of p-local Mackey algebras.. 2013. ⟨hal-00819356v2⟩