Weights in a Benson-Solomon block

TL;DR

This paper investigates the hypothetical weights in Benson-Solomon exotic fusion systems, establishing a unique associated pair and determining the number of weights as 12, independent of the field of definition.

Contribution

It proves the uniqueness of the pair of fusion system and cohomology classes for Benson-Solomon systems and calculates the number of weights as 12.

Findings

Number of weights in Benson-Solomon blocks is 12.

Unique pair of fusion system and cohomology classes exists for Benson-Solomon systems.

Explicit listing of centric radical subgroups and automorphism groups was performed.

Abstract

To each pair consisting of a saturated fusion system over a $p$-group together with a compatible family of K\"ulshammer-Puig cohomology classes, one can count weights in a hypothetical block algebra arising from these data. When the pair arises from a bonafide block of a finite group algebra in characteristic $p$, the number of conjugacy classes of weights is supposed to be the number of simple modules in the block. We show that there is unique such pair associated with each Benson-Solomon exotic fusion system, and that the number of weights in a hypothetical Benson-Solomon block is $12$, independently of the field of definition. This is carried out in part by listing explicitly up to conjugacy all centric radical subgroups and their outer automorphism groups in these systems.