Weight conjectures for fusion systems

TL;DR

This paper explores local-to-global conjectures in the representation theory of finite groups, focusing on fusion systems and their cohomology classes, motivated by block theory conjectures.

Contribution

It formulates and begins investigating new local conjectures for fusion systems and associated cohomology data, extending ideas from block theory.

Findings

Proposes new local conjectures for fusion systems.

Establishes a framework linking fusion systems and cohomology.

Initiates analysis of conjectures in the context of arbitrary triples.

Abstract

Many of the conjectures of current interest in the representation theory of finite groups in characteristic $p$ are local-to-global statements, in that they predict consequences for the representations of a finite group $G$ given data about the representations of the $p$-local subgroups of $G$. The local structure of a block of a group algebra is encoded in the fusion system of the block together with a compatible family of K\"ulshammer-Puig cohomology classes. Motivated by conjectures in block theory, we state and initiate investigation of a number of seemingly local conjectures for arbitrary triples $(S,\mathcal{F},\alpha)$ consisting of a saturated fusion system $\mathcal{F} $ on a finite $p$-group $S$ and a compatible family $\alpha$.