A 3D Spinorial View of 4D Exceptional Phenomena

  • Pierre-Philippe DechantEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 159)


We discuss a Clifford algebra framework for discrete symmetry groups (such as reflection, Coxeter, conformal and modular groups), leading to a surprising number of new results. Clifford algebras allow for a particularly simple description of reflections via ‘sandwiching’. This extends to a description of orthogonal transformations in general by means of ‘sandwiching’ with Clifford algebra multivectors, since all orthogonal transformations can be written as products of reflections by the Cartan-Dieudonné theorem. We begin by viewing the largest non-crystallographic reflection/Coxeter group \(H_4\) as a group of rotations in two different ways—firstly via a folding from the largest exceptional group \(E_8\), and secondly by induction from the icosahedral group \(H_3\) via Clifford spinors. We then generalise the second way by presenting a construction of a 4D root system from any given 3D one. This affords a new, spinorial, perspective on 4D phenomena, in particular as the induced root systems are precisely the exceptional ones in 4D, and their unusual automorphism groups are easily explained in the spinorial picture; we discuss the wider context of Platonic solids, Arnold’s trinities and the McKay correspondence. The multivector groups can be used to perform concrete group-theoretic calculations, e.g. those for \(H_3\) and \(E_8\), and we discuss how various representations can also be constructed in this Clifford framework; in particular, representations of quaternionic type arise very naturally.


Root System Simple Root Clifford Algebra Dynkin Diagram Coxeter Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



I would like to thank Reidun Twarock, Anne Taormina, David Hestenes, Anthony Lasenby, John Stillwell, Jozef Siran, Robert Wilson and Ben Fairbairn.


  1. 1.
    Vladimir Igorevich Arnold. Symplectization, complexification and mathematical trinities. The Arnoldfest, pages 23-37, 1999.Google Scholar
  2. 2.
    Vladimir Igorevich Arnold. Mathematics: Frontiers and perspectives. Amer Mathematical Society, 2000.Google Scholar
  3. 3.
    Nicolas Bourbaki. Groupes et algèbres de Lie, chapitres 4, 5 et 6. Masson, Paris, 1981.Google Scholar
  4. 4.
    T. Damour, M. Henneaux, and H. Nicolai. \(E_{10}\) and a ‘small tension expansion’ of M-Theory. Physical Review Letters, 89:221601, 2002.Google Scholar
  5. 5.
    Pierre-Philippe Dechant. Models of the Early Universe. PhD thesis, University of Cambridge, UK, 2011.Google Scholar
  6. 6.
    Pierre-Philippe Dechant. Clifford algebra unveils a surprising geometric significance of quaternionic root systems of Coxeter groups. Advances in Applied Clifford Algebras, 23(2):301-321, 2013, doi: 10.1007/s00006-012-0371-3.Google Scholar
  7. 7.
    Pierre-Philippe Dechant. Platonic solids generate their four-dimensional analogues. Acta Crystallographica Section A: Foundations of Crystallography, 69(6):592-602, 2013.Google Scholar
  8. 8.
    Pierre-Philippe Dechant. A Clifford algebraic framework for Coxeter group theoretic computations. Advances in Applied Clifford Algebras, 24(1):89-108, 2014.Google Scholar
  9. 9.
    Pierre-Philippe Dechant. Clifford algebra is the natural framework for root systems and Coxeter groups. group theory: Coxeter, conformal and modular groups. Advances in Applied Clifford Algebras, 2015, doi: 10.1007/s00006-015-0584-3.Google Scholar
  10. 10.
    Pierre-Philippe Dechant. Rank-3 root systems induce root systems of rank 4 via a new Clifford spinor construction. Journal of Physics: Conference Series, 597(1):012027, 2015.Google Scholar
  11. 11.
    Pierre-Philippe Dechant. The birth of \(E_8\) out of the (s)pinors of the icosahedron submitted to Proceedings of the Royal Society A 20150504, 2016, doi: 10.1098/rspa.2015.0504.Google Scholar
  12. 12.
    Pierre-Philippe Dechant. The E \(_{8}\) geometry from a Clifford perspective, Advances in Applied Clifford Algebras, 2016.Google Scholar
  13. 13.
    Pierre-Philippe Dechant, Céline Boehm, and Reidun Twarock. Novel Kac-Moody-type affine extensions of non-crystallographic Coxeter groups. Journal of Physics A: Mathematical and Theoretical, 45(28):285202, 2012.Google Scholar
  14. 14.
    Pierre-Philippe Dechant, Céline Boehm, and Reidun Twarock. Affine extensions of noncrystallographic Coxeter groups induced by projection. Journal of Mathematical Physics, 54(9), 2013.Google Scholar
  15. 15.
    Pierre-Philippe Dechant, Jess Wardman, Tom Keef, and Reidun Twarock. Viruses and fullerenes—symmetry as a common thread? Acta Crystallographica Section A, 70(2):162–167, Mar 2014.Google Scholar
  16. 16.
    Chris Doran and Anthony N. Lasenby. Geometric Algebra for Physicists. Cambridge University Press, Cambridge, 2003.Google Scholar
  17. 17.
    Tohru Eguchi, Hirosi Ooguri, and Yuji Tachikawa. Notes on the K3 surface and the Mathieu group \(M_{24}\). Experimental Mathematics, 20(1):91-96, 2011.Google Scholar
  18. 18.
    Tohru Eguchi, Yuji Sugawara, and Anne Taormina. Liouville field, modular forms and elliptic genera. Journal of high energy physics, 2007(03):119, 2007.Google Scholar
  19. 19.
    Terry Gannon. Moonshine beyond the Monster: The bridge connecting algebra, modular forms and physics. Cambridge University Press, 2006.Google Scholar
  20. 20.
    David J. Gross, Jeffrey A. Harvey, Emil J. Martinec, and Ryan Rohm. Heterotic String Theory. 1. The Free Heterotic String. Nucl.Phys., B256:253, 1985.Google Scholar
  21. 21.
    M. Henneaux, D. Persson, and P. Spindel. Spacelike Singularities and Hidden Symmetries of Gravity. Living Reviews in Relativity, 11:1-+, April 2008.Google Scholar
  22. 22.
    David Hestenes. Space-Time Algebra. Gordon and Breach, New York, 1966.Google Scholar
  23. 23.
    David Hestenes. New foundations for classical mechanics; 2nd ed. Fundamental theories of physics. Kluwer, Dordrecht, 1999.Google Scholar
  24. 24.
    David Hestenes and Garret Sobczyk. Clifford algebra to geometric calculus: a unified language for mathematics and physics. Fundamental theories of physics. Reidel, Dordrecht, 1984.Google Scholar
  25. 25.
    M. Koca, M. Al-Barwani, and R. Koç. Quaternionic root systems and subgroups of the Aut(\(\text{ F }_{4}\)). Journal of Mathematical Physics, 47(4):043507-+, April 2006.Google Scholar
  26. 26.
    M. Koca, R. Koç, and M. Al-Barwani. Quaternionic roots of SO(8), SO(9), \(F_{4}\) and the related Weyl groups. Journal of Mathematical Physics, 44:3123-3140, July 2003.Google Scholar
  27. 27.
    Mehmet Koca, Ramazan Koc, and Muataz Al-Barwani. Noncrystallographic Coxeter group \(H_4\) in \(E_8\). Journal of Physics A: Mathematical and General, 34(50):11201, 2001.Google Scholar
  28. 28.
    John McKay. Graphs, singularities, and finite groups. In Proc. Symp. Pure Math, volume 37, pages 183-186, 1980.Google Scholar
  29. 29.
    R. V. Moody and J. Patera. Quasicrystals and icosians. Journal of Physics A: Mathematical and General, 26(12):2829, 1993.Google Scholar
  30. 30.
    A. N. Schellekens. Introduction to Conformal Field Theory. Fortschritte der Physik, 44:605–705, 1996.Google Scholar
  31. 31.
    O. P. Shcherbak. Wavefronts and reflection groups. Russian Mathematical Surveys, 43(3):149, 1988.Google Scholar
  32. 32.
    Anne Taormina and Katrin Wendland. A twist in the \(M_{24}\) moonshine story. arXiv preprintarXiv:1303.3221, 2013.Google Scholar
  33. 33.
    R. A. Wilson. Geometriae Dedicata, 20:157, 1986.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of YorkYorkUK

Personalised recommendations