Reduced, tame and exotic fusion systems (Q2901907)

This paper reports on a program to search for and understand exotic fusion systems. These are fusion systems that are not the fusion systems of any finite group. If \(G\) is a finite group and \(S\) a Sylow \(p\)-subgroup of \(G\) then the fusion category of \(G\) is a category whose objects are the subgroups of \(S\) and whose morphisms encode the conjugacy relations between these subgroups.NEWLINENEWLINEThe authors define notions of \textit{tame} and \textit{reduced} fusion systems. A tame system is essentially one which is realized by a finite group for which all automorphisms of the fusion system are induced by automorphisms of \(G\). The authors describe a canonical reduction of a fusion system and show that for every saturated system over a finite \(p\)-group if the reduction is tame then the original was tame. Thus it is only necessary to consider reduced systems to find exotic fusion systems. Further the authors show how to decompose reduced fusion systems into products. A final section of the paper gives detailed examples intended to illustrate how the techniques given in the paper can be used to prove the tameness of certain reduced fusion systems.