On the local rank of fusion systems

TL;DR

This paper introduces the concept of local rank in fusion systems to reformulate Alperin's weight conjecture within block fusion systems, building on prior work by Kn"orr and Robinson.

Contribution

It defines local rank for fusion systems and applies it to reformulate a major conjecture in block theory, offering a new perspective.

Findings

Reformulation of Alperin's weight conjecture using local rank

Introduction of the local rank concept for fusion systems

Connection to existing work by Kn"orr and Robinson

Abstract

In this paper we define the notion of local rank for fusion systems so as to reformulate the Alperin's weight conjecture in the framework of block fusion systems following the work by Kn\"orr and Robinson.