On blocks stably equivalent to a quantum complete intersection of dimension 9 in characteristic 3 and a case of the abelian defect group conjecture

TL;DR

This paper proves Broue's abelian defect group conjecture for certain 3-blocks with defect 2 using stable equivalences and automorphism group isomorphisms, advancing understanding in modular representation theory.

Contribution

It demonstrates the conjecture for specific blocks by leveraging stable equivalences of Morita type and automorphism group analysis.

Findings

Broue's conjecture holds for 3-blocks of defect 2 with a unique simple module in the Brauer correspondent.

Stable equivalences induce isomorphisms between automorphism groups of self-injective algebras.

The approach confirms the conjecture in a new class of blocks using Rouquier's stable equivalence techniques.

Abstract

Using a stable equivalence due to Rouquier, we prove that Broue's abelian defect group conjecture holds for 3-blocks of defect 2 whose Brauer correspondent has a unique isomorphism class of simple modules. The proof makes use of the fact, also due to Rouquier, that a stable equivalence of Morita type between self-injective algebras induces an isomorphism between the connected components of the outer automorphism groups of the algebras.