Description
A Clifford algebraic approach to reflection groups and root systems 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 CartanDieudonne 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.Period  6 Aug 2017 → 12 Aug 2017 

Event title  Yau Institute: Tsinghua Summer Workshop in Geometry and Physics 2017 
Event type  Conference 
Location  Beijing, ChinaShow on map 
Keywords
 geometry
 physics
 string theory
 mathematical physics
 calabiyau
 Clifford algebras
 Coxeter groups
 root systems
 Lie theory
 representation theory
 group theory
 quaternions
 Coxeter plane
 exceptional geometries
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Related content

Research output

The birth of E8 out of the spinors of the icosahedron
Research output: Contribution to journal › Article › peerreview

The E8 geometry from a Clifford perspective
Research output: Contribution to journal › Article › peerreview

Rank3 root systems induce root systems of rank 4 via a new Clifford spinor construction
Research output: Contribution to journal › Article › peerreview

Clifford Algebra Unveils a Surprising Geometric Significance of Quaternionic Root Systems of Coxeter Groups
Research output: Contribution to journal › Article › peerreview

A Clifford Algebraic Framework for Coxeter Group Theoretic Computations
Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

A 3D spinorial view of 4D exceptional phenomena
Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

A Clifford algebraic construction of the E8 root system
Research output: Contribution to conference › Paper › peerreview

Clifford algebra is the natural framework for root systems and Coxeter groups: Group theory: Coxeter, conformal and modular groups
Research output: Contribution to journal › Article › peerreview