Equivalences between blocks of cohomological Mackey algebras

TL;DR

This paper establishes a connection between derived equivalences of blocks in group algebras and their corresponding blocks in cohomological Mackey algebras, providing new insights into Broué's conjecture.

Contribution

It proves that permeable derived equivalences between group algebra blocks induce derived equivalences between their cohomological Mackey algebra blocks.

Findings

Permeable derived equivalences imply Mackey algebra block equivalences.

Splendid derived equivalences in group algebras lead to Mackey algebra block equivalences.

Supports Broué's abelian defect group conjecture in the Mackey algebra context.

Abstract

Let $G$ be a finite group and $(K,\mathcal{O},k)$ be a $p$-modular system "large enough". Let $R=\mathcal{O}$ or $k$. There is a bijection between the blocks of the group algebra $RG$ and the central primitive idempotents (the blocks) of the so-called cohomological Mackey algebra $co\mu_{R}(G)$. Here, we prove that a so-called permeable derived equivalence between two blocks of group algebras implies the existence of a derived equivalence between the corresponding blocks of cohomological Mackey algebras. In particular, in the context of Brou\'e's abelian defect group conjecture, if two blocks are splendidly derived equivalent, then the corresponding blocks of cohomological Mackey algebras are derived equivalent.