Periodic Reporting for period 1 - FusionSystems (Simple fusion systems and linking systems)
Reporting period: 2016-08-03 to 2018-08-02
The theory of fusion systems provides one context for such a simplification. A fusion system is considered a snapshot of a finite group at a prime number p, although certain ""exotic"" fusion systems provide snapshots of finite simple groups that do not actually exist. Fusion systems were first considered in the study of representations of groups, namely the study of the ways in which groups can realize their symmetries in nature. They were later found to be important in certain segments of topology, or ""rubber-sheet geometry"". More recently, and because certain technical difficulties in the CFSG do not arise in fusion systems, the study of fusion systems has been taken up as an avenue to simplify the CFSG.
The Dichotomy Theorem says that fusion systems can be partitioned into those of component type and those of characteristic p-type. The objective of the FusionSystems project was to make significant contributions to the classification of simple fusion systems of component type at the prime two, as well as the classification of simple fusion systems of characteristic p-type for an arbitrary prime p. The action will result in the publication of three peer-reviewed articles in international journals and/or conference proceedings, including public access to final-draft post-refereed versions of the articles. In addition, the action results in thirteen conference and seminar talks communicating the results of the action and of closely related research, one talk to the general public, and the support of four international visitors.
A classification of fusion systems with involution centralizer having a Benson-Solomon component, the subject of the second manuscript, is broadly awaited by researchers in fusion systems. Because a Benson-Solomon system is not the fusion system of any finite group, any simple fusion system having such an involution centralizer would necessarily be a new exotic system at the prime 2, which has been sought by many authors. The FusionSystems project's results provide additional evidence there is no such exotic system, strengthening the case for a conjecture of R. Solomon. The results also form part of one of the major steps in the classification of simple fusion systems of component type, namely the proof of an analogue of a theorem of J. Walter within the CFSG.
The description of the centric and radical subgroups in the Benson-Solomon systems builds on previous research on these systems. It allowed FusionSystems in the third manuscript to compute the number of simple representations that a Benson-Solomon finite simple group would have if it existed, thus answering a question posed by one of the leading figures in the representation theory of finite groups.