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On Morita equivalences with endopermutation source and isotypies

Xin Huang

TL;DR

The paper defines almost isotypy as a natural relaxation between Broué's weak isotypy and Linckelmann's isotypy, for blocks sharing a defect group $P$ and fusion system $\\mathcal{F}$. It shows that a Morita equivalence realized by a bimodule with an endopermutation source $V$ yields an almost isotypy whenever the source character values $\\rho_V$ are integers; under $p \ge 3$ and abelian $P$, signs can be chosen to obtain an actual isotypy in Lin18b. The results connect local Morita equivalences induced by slashed modules with global decomposition numbers, and provide a lifting theory from modular to integral settings, illustrating how endopermutation sources govern fusion-compatible correspondences between blocks. A counterexample for $p=2$ and $P=Q_8$ shows that almost isotypy is the optimal notion in general, even when $\\rho_V$ is integral. Together with prior work by Kessar–Linckelmann and Lin18b, the paper advances understanding of how endopermutation sources control isotypy-type relations in block theory and Morita theory.

Abstract

We introduce a new type of equivalence between blocks of finite group algebras called an almost isotypy. An almost isotypy restricts to a weak isotypy in Broué's original definition, and it is slightly weaker than Linckelmann's version. We show that a bimodule of two block algebras of finite groups - which has an endopermutation module as a source and which induces a Morita equivalence - gives rise, via slash functors, to an almost isotypy if the character values of a (hence any) source are rational integers. Consequently, if two blocks are Morita equivalent via a bimodule with endopermutation source, then they are almost isotypic. We also explain why the notion of almost isotypies is reasonable.

On Morita equivalences with endopermutation source and isotypies

TL;DR

The paper defines almost isotypy as a natural relaxation between Broué's weak isotypy and Linckelmann's isotypy, for blocks sharing a defect group and fusion system . It shows that a Morita equivalence realized by a bimodule with an endopermutation source yields an almost isotypy whenever the source character values are integers; under and abelian , signs can be chosen to obtain an actual isotypy in Lin18b. The results connect local Morita equivalences induced by slashed modules with global decomposition numbers, and provide a lifting theory from modular to integral settings, illustrating how endopermutation sources govern fusion-compatible correspondences between blocks. A counterexample for and shows that almost isotypy is the optimal notion in general, even when is integral. Together with prior work by Kessar–Linckelmann and Lin18b, the paper advances understanding of how endopermutation sources control isotypy-type relations in block theory and Morita theory.

Abstract

We introduce a new type of equivalence between blocks of finite group algebras called an almost isotypy. An almost isotypy restricts to a weak isotypy in Broué's original definition, and it is slightly weaker than Linckelmann's version. We show that a bimodule of two block algebras of finite groups - which has an endopermutation module as a source and which induces a Morita equivalence - gives rise, via slash functors, to an almost isotypy if the character values of a (hence any) source are rational integers. Consequently, if two blocks are Morita equivalent via a bimodule with endopermutation source, then they are almost isotypic. We also explain why the notion of almost isotypies is reasonable.
Paper Structure (8 sections, 32 theorems, 72 equations)

This paper contains 8 sections, 32 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On character values of endopermutation modules
  4. On local Morita equivalences induced by slashed modules
  5. Lifting local Morita equivalences from $k$ to $\mathcal{O}$
  6. On generalised decomposition numbers
  7. Proof of Theorem \ref{['theo:main']}
  8. Proof of Proposition \ref{['theo:counterexample']}

Key Result

Theorem 1.4

Let $V$ be an indecomposable $\mathcal{F}$-stable endopermutation $\mathcal{O} P$-module, viewed as an $\mathcal{O}\Delta P$-module via the canonical isomorphism $\Delta P\cong P$. Let $M$ be an indecomposable direct summand of the $\mathcal{O} Gb$-$\mathcal{O} Hc$-bimodule Suppose that $M$ induces a Morita equivalence between $\mathcal{O} Gb$ and $\mathcal{O} Hc$. For any $1\neq Q\subseteq P$, l

Theorems & Definitions (40)

  • Definition 1.2: cf. Broue1990
  • Definition 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Remark 1.6
  • Theorem 1.7
  • Lemma 2.3
  • Proposition 2.6: Lin08
  • Lemma 2.7: a variation of Lin08
  • Theorem 2.9: Dade's slashed modules; see e.g. Lin18b
  • ...and 30 more