Activity: Talk or presentation › Workshop

Pierre-Philippe Dechant - Keynote/plenary speaker

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 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.

6 Aug 2017 → 12 Aug 2017

Title | Yau Institute: Tsinghua Summer Workshop in Geometry and Physics 2017 |
---|---|

Period | 6/08/17 → 12/08/17 |

Location | Yau Institute, Tsinghua University |

City | Beijing |

Country | China |

Degree of recognition | International event |

- geometry, physics, string theory, mathematical physics, calabi-yau, Clifford algebras, Coxeter groups, root systems, Lie theory, representation theory, group theory, quaternions, Coxeter plane, exceptional geometries

## The E8 geometry from a Clifford perspective

Research output: Contribution to journal › Article

## 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

## The birth of E8 out of the spinors of the icosahedron

Research output: Contribution to journal › Article

## A 3D spinorial view of 4D exceptional phenomena

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

## Rank-3 root systems induce root systems of rank 4 via a new Clifford spinor construction

Research output: Contribution to journal › Article

## A Clifford Algebraic Framework for Coxeter Group Theoretic Computations

Research output: Contribution to journal › Article

## A Clifford algebraic construction of the E8 root system

Research output: Contribution to conference › Paper

## Clifford Algebra Unveils a Surprising Geometric Significance of Quaternionic Root Systems of Coxeter Groups

Research output: Contribution to journal › Article

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