Advances in Applied Clifford Algebras

, Volume 27, Issue 1, pp 17–31 | Cite as

Clifford Algebra is the Natural Framework for Root Systems and Coxeter Groups. Group Theory: Coxeter, Conformal and Modular Groups

Article

Abstract

In this paper, we make the case that Clifford algebra is the natural framework for root systems and reflection groups, as well as related groups such as the conformal and modular groups: The metric that exists on these spaces can always be used to construct the corresponding Clifford algebra. Via the Cartan–Dieudonné theorem all the transformations of interest can be written as products of reflections and thus via ‘sandwiching’ with Clifford algebra multivectors. These multivector groups can be used to perform concrete calculations in different groups, e.g. the various types of polyhedral groups, and we treat the example of the tetrahedral group A 3 in detail. As an aside, this gives a constructive result that induces from every 3D root system a root system in dimension four, which hinges on the facts that the group of spinors provides a double cover of the rotations, the space of 3D spinors has a 4D euclidean inner product, and with respect to this inner product the group of spinors can be shown to be closed under reflections. In particular the 4D root systems/Coxeter groups induced in this way are precisely the exceptional ones, with the 3D spinorial point of view also explaining their unusual automorphism groups. This construction simplifies Arnold’s trinities and puts the McKay correspondence into a wider framework. We finally discuss extending the conformal geometric algebra approach to the 2D conformal and modular groups, which could have interesting novel applications in conformal field theory, string theory and modular form theory.

Mathematics Subject Classification

Primary 51F15 20F55 Secondary 15A66 52B15 

Keywords

Clifford algebras Coxeter groups Root systems Group theory Representations Spinors Binary polyhedral groups Exceptional phenomena Trinities McKay correspondence Conformal group Modularity 

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References

  1. 1.
    Arnold, VI.: Symplectization, Complexification and Mathematical Trinities. The Arnoldfest, pp. 23–37 (1999)Google Scholar
  2. 2.
    Arnold, VI.: Mathematics: Frontiers and Perspectives. American Mathematical Society (2000)Google Scholar
  3. 3.
    Dechant, P.-P.: Models of the Early Universe. Ph.D. thesis, University of Cambridge, UK, 2011Google Scholar
  4. 4.
    Dechant, P.-P.: Clifford algebra unveils a surprising geometric significance of quaternionic root systems of Coxeter groups. Adv. Appl. Clifford Algebr. 23(2), 301–321 (2013). doi: 10.1007/s00006-012-0371-3
  5. 5.
    Dechant P.-P.: Platonic solids generate their four-dimensional analogues. Acta Crystallogr. Sect. A: Found. Crystallogr. 69(6), 592–602 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dechant P.-P.: A Clifford algebraic framework for Coxeter group theoretic computations. Adv. Appl. Clifford Algebr. 24(1), 89–108 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dechant, P.-P.: Rank-3 root systems induce root systems of rank 4 via a new Clifford spinor construction. J. Phys.: Conf. Ser.; Proc. Group 30, 2015Google Scholar
  8. 8.
    Dechant P.-P., Boehm C., Twarock R.: Novel Kac-Moody-type affine extensions of non-crystallographic Coxeter groups. J. Phys. A: Math. Theor. 45(28), 285202 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Dechant, P.-P.: Céline Boehm, and Reidun Twarock. Affine extensions of non-crystallographic Coxeter groups induced by projection. J. Math. Phys. 54(9) (2013)Google Scholar
  10. 10.
    Doran C., Lasenby A.N.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge (2003)CrossRefMATHGoogle Scholar
  11. 11.
    Eguchi T., Ooguri H., Tachikawa Y.: Notes on the K3 surface and the Mathieu group M 24. Exp. Math. 20(1), 91–96 (2011)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Eguchi T., Taormina A.: Unitary representations of the \({\mathcal{N}=4}\) superconformal algebra. Phys. Lett. B 196(1), 75–81 (1987)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Eguchi T., Taormina A.: Character formulas for the \({\mathcal{N}=4}\) superconformal algebra. Phys. Lett. B 200(3), 315–322 (1988)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Gannon, T.: Moonshine beyond the Monster: The bridge connecting algebra, modular forms and physics. Cambridge University Press, 2006Google Scholar
  15. 15.
    Hestenes, D.: Space-Time Algebra. Gordon and Breach, New York, 1966Google Scholar
  16. 16.
    Hestenes, D.: New Foundations for Classical Mechanics; 2nd ed. Fundamental Theories of Physics. Kluwer, Dordrecht, 1999Google Scholar
  17. 17.
    Hestenes, D.: Point Groups and Space Groups in Geometric Algebra, pp. 3–34. Birkhäuser, Boston, 2002Google Scholar
  18. 18.
    Hestenes D., Holt J.W.: The crystallographic space groups in geometric algebra. J. Math. Phys. 48, 023514 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics. Fundamental Theories of Physics. Reidel, Dordrecht, 1984Google Scholar
  20. 20.
    Hitzer E., Perwass C.: Interactive 3D space group visualization with CLUCalc and the Clifford Geometric Algebra description of space groups. Adv. Appl. Clifford Algebr. 20, 631–658 (2010). doi: 10.1007/s00006-010-0214-z
  21. 21.
    Humphreys J.E.: Reflection groups and Coxeter groups. Cambridge University Press, Cambridge (1990)CrossRefMATHGoogle Scholar
  22. 22.
    Koca, M., Al-Barwani, M., Koç, R.: Quaternionic root systems and subgroups of the Aut(F4). J. Math. Phys. 47(4):043507 (2006)Google Scholar
  23. 23.
    Moody R.V., Patera J.: Quasicrystals and icosians. J. Phys. A: Math. General 26(12), 2829 (1993)ADSMathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Ooguri H., Petersen J.L., Taormina A.: Modular invariant partition functions for the doubly extended \({\mathcal{N}=4}\) superconformal algebras. Nuclear Phys. B 368(3), 611–624 (1992)ADSMathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Petersen J.L., Taormina A.: Modular properties of doubly extended \({\mathcal{N}=4}\) superconformal algebras and their connection to rational torus models (i). Nuclear Phys. B 354(2), 689–710 (1991)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Petersen J.L., Taormina A.: Characters of the \({\mathcal{N}=4}\) superconformal algebra with two central extensions (ii). massless representations. Nuclear Phys. B 333(3), 833–854 (1990)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Taormina, A., Wendland, K.: A twist in the M 24 moonshine story. arXiv preprint arXiv:1303.3221, 2013
  28. 28.
    Wiles, A.: Modular elliptic curves and Fermat’s last theorem. Ann. Math. pp. 443–551 (1995)Google Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of YorkYorkUK

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