Reduced, tame and exotic fusion systems

TL;DR

This paper introduces reduced and tame fusion systems to better understand exotic fusion systems, showing how all saturated fusion systems relate to these new classes and their realizability.

Contribution

It defines reduced and tame fusion systems, establishing their roles in classifying and understanding exotic fusion systems and their relation to realizability.

Findings

Every saturated fusion system reduces to a reduced fusion system.

A reduced fusion system is tame if and only if it is realizable.

Non-tame reduced fusion systems are reductions of exotic, nonrealizable fusion systems.

Abstract

We define here two new classes of saturated fusion systems, reduced fusion systems and tame fusion systems. These are motivated by our attempts to better understand and search for exotic fusion systems: fusion systems which are not the fusion systems of any finite group. Our main theorems say that every saturated fusion system reduces to a reduced fusion system which is tame only if the original one is realizable, and that every reduced fusion system which is not tame is the reduction of some exotic (nonrealizable) fusion system.