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Hyperfocal subalgebras of hyperfocal abelian Frobenius blocks

Xueqin Hu, Kun Zhang, Yuanyang Zhou

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.

Hyperfocal subalgebras of hyperfocal abelian Frobenius blocks

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.
Paper Structure (8 sections, 46 theorems, 45 equations)

This paper contains 8 sections, 46 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperfocal abelian Frobenius blocks
  4. The bimodule structure and the Brauer quotients of $\mathbb{A}_\delta$
  5. A stable equivalence of Morita type
  6. Applications
  7. Klein four group case
  8. Cyclic group case

Key Result

Lemma 3.3

Let $Q$ be a $p$-subgroup and $C$ an indecomposable relatively $\hbox{$\mathcal{O}$} Q$-separable $Q$-interior algebra. Assume that $C$ has a unique local pointed group $Q_\epsilon$ and as $\hbox{$\mathcal{O}$}(Q\times Q)$-modules, the indecomposable direct summand of $C$ has the form $\mathrm{Ind}_

Theorems & Definitions (89)

  • Lemma 3.3
  • proof
  • Corollary 3.4
  • proof
  • Lemma 3.6
  • proof
  • Proposition 3.7
  • proof
  • Remark 3.9
  • Proposition 3.12
  • ...and 79 more