Reflection Groups (MAGIC001)
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Let V be a Euclidean space. The finite reflection groups on V play a central role in the study of finite groups and of algebraic groups. We shall begin by classifying all the finite subgroup of the orthogonal group O(V) when V has dimension 2 or 3. For G a finite subgroup of O(V), we then introduce fundamental regions for the action of G on V. Following this we define Coxeter groups as finite groups generated by reflections in O(V) which act effectively on V. To study such subgroups of O(V) we introduce root systems and show that G simply transitively on the positive systems in the root system. In the final chapter, we classify root systems and thus also classify the Coxeter groups. This classification is as usual parameterized by the Coxeter diagrams. This classification is as usual parameterized by the Coxeter diagrams. As time allows I will cover further material. This will be chosen from: Presentations of Coxeter Groups, Invariants of Coxeter Groups, Affine Reflection groups, Complex reflection groups.
Spring 2010 (Monday, January 11 to Friday, March 19)
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