Hyperfocal subalgebras of hyperfocal abelian Frobenius blocks

TL;DR

This paper introduces hyperfocal abelian Frobenius blocks, establishing stable Morita equivalences and analyzing their structures, thereby advancing understanding of blocks with specific hyperfocal subgroups.

Contribution

It defines hyperfocal abelian Frobenius blocks and proves stable Morita equivalences with group algebras, extending knowledge on block structures and Broue's conjecture.

Findings

Stable Morita equivalence between hyperfocal subalgebras and Frobenius group algebras.

Partial description of block structures with Klein four and cyclic hyperfocal subgroups.

Verification of Broue's abelian defect group conjecture for blocks with Klein four hyperfocal subgroups.

Abstract

In this paper, we introduce a class of blocks which is called hyperfocal abelian Frobenius blocks.This class of blocks is an analogous version of the block with abelian defect group and Frobenius inertial quotient at hyperfocal level and includes the blocks with Klein four hyperfocal subgroups and cyclic hyperfocal subgroups. We show that there is a stable equivalence of Morita type between the hyperfocal subalgebras of the hyperfocal abelian Frobenius blocks and a group algebra of a Frobenius group associated with the hyperfocal subgroup of the block. As applications, we can partially describe some structures of the blocks with Klein four hyperfocal subgroups and cyclic hyperfocal subgroups,such as the structures of their hyperfocal subalgebras in terms of derived categories and the structures of their characters. As a consequence, we show that Broue's abelian defect group conjecture holds for blocks with Klein four hyperfocal subgroups.