A Clifford algebraic approach to reflection groups and root systems
Student No.:50
Time:16:45-17:45 Thu 2017-8-10
Instructor:Pierre Dechant  
Place:Floor 3, Jin Chun Yuan West Building
Starting Date:2017-8-10
Ending Date:2017-8-10

Abstract: Reflection symmetries are ubiquitous in mathematics and science, ranging from their central role in Lie groups and algebras, gravitational and particle physics, to the structure of viruses and fullerenes. The mathematical properties of reflection groups are conveniently encoded in their root systems. I argue that Clifford algebras are a useful framework for such reflection groups and root systems for four reasons: firstly, due to the uniquely simple formula for reflections that Clifford algebras provide, which actually provides a double cover; secondly, the connection between this spin double cover and root systems; thirdly, the emergence of various  geometric objects such as (collections of) planes that satisfy complex or quaternionic relations. Finally, the Cartan-Dieudonne theorem opens up this approach to a wide class of symmetry groups, which can be expressed as products of reflections such as orthogonal, conformal and modular groups. This more geometric Clifford framework is thus a very general approach to group and representation theory and has already yielded new insights into exceptional geometries (such as D_4, F_4, H_4, E_8), ADE correspondences, quaternionic representations, modular and braid groups, as well as the geometry of the Coxeter plane and element.