A Clifford Algebraic Framework for Coxeter Group Theoretic Computations

Pierre-Philippe Dechant
  • Advances in Applied Clifford Algebras, November 2013, Springer Science + Business Media
  • DOI: 10.1007/s00006-013-0422-4

A new approach to reflections

What is it about?

Coxeter groups are a mathematical framework for describing reflections. Clifford algebra has a uniquely simple way of performing reflections. Clifford algebra does not seem to have been applied to the Coxeter group and root system framework. The combination of both paradigms resulted in a number of surprising results. (This paper won the conference prize at AGACSE 2012.)

Why is it important?

These include an understanding of 4D geometry through rotations (spinors) in 3D. Since there is an intimate connection between the conformal group (including translations) in n dimensions and the orthogonal group in n+2 dimensions, lattices can be generated multiplicatively in this framework; we showed that a conformal framework is appealing for root systems and also quasilattices. We furthermore provide a geometric interpretation of complex eigenvalues of the Coxeter element.

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The following have contributed to this page: Pierre-Philippe Dechant