Fusion systems in representation theory

TL;DR

This paper introduces fusion systems, exploring their origins in classical group theory, developing their abstract theory, and applying them to problems in representation theory, without discussing topological applications.

Contribution

It presents an overview of fusion systems, their theoretical development, and their applications to representation theory, filling a gap in the literature.

Findings

Fusion systems generalize classical fusion phenomena.

Applications to representation theory are demonstrated.

The notes exclude topological applications.

Abstract

These notes arose from three lectures on fusion system I gave at the University of Valencia in February 2023. We first consider fusion phenomena in classical group theory. Then we develop the abstract theory of fusion systems. Finally applications to representation theory are given. We will not consider applications in topology.