Coxeter Groups: Reflections and Fundamental Regions

This lecture covers the concept of Coxeter groups, focusing on reflections and fundamental regions. It explains how any reflection in a Coxeter group is conjugate to a simple reflection, using examples like the rigid symmetries of a cube. The lecture also delves into the fundamental regions of Coxeter groups, demonstrating how they are generated by reflections with respect to walls. The classification of Coxeter groups by Coxeter graphs is discussed, along with the properties of simple roots and fundamental weights. Various examples are provided to illustrate these concepts.