Abstract
Associated with the real Clifford algebra Cl(p, q), p + q = n, is the finite Dirac group G(p, q) of order 2n+1. The quotient group V n = G(p, q)/ *#x007B;± 1*#x007D;, viewed additively, is an ndimensional vector space over GF(2) = *#x007B;0, 1*#x007D; which comes equipped with a quadratic form Q and associated alternating bilinear form B. Properties of the finite geometry over GF(2) of V n B, Q — in part familiar, in part less so — are given a rather full description, and a dictionary of translation into their Dirac group counterparts is provided. The knowledge gained is used, in conjunction with facts concerning representations of G(p, q), to give a pleasantly clean derivation of the well-known table of Porteous (1969/1981) of the algebras Cl(p, q). In particular the finite geometry highlights the “antisymmetry” of the table about the column p - q = -1. Several low-dimensional illustrations are given of the application of finite geometry results to Dirac groups. Particular emphasis is laid on certain interesting finite geometry symmetry methods, the latter being given a rather full treatment in the appendices. Finite geometry is also used to study the automorphisms of the Dirac groups, and the splitting of certain exact sequences.
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References
Artin, E. (1957), ‘Geometric Algebral’. Interscience, New York.
Benn, I.M. & Tucker, R.W. (1987), ‘An Introduction to Spinors and Geometryl’. Adam Hilger, Bristol.
Braden, H.W. (1985), ‘N-dimensional spinors: their properties in terms of finite groupsl’. J. Math. Phys. 26, pp. 613–620.
Burnside, W. (1911), ‘Theory of Groups of Finite Orderl’. Cambridge University Press, Cambridge.
Cameron, P.J. & Van Lint, J.H. (1991), ‘Designs, Graphs, Codes and their Linksl’. Cambridge University Press, Cambridge.
Cartan, E. (1908) (exposé d’après l’article allemand de E. Study), ‘Nombres Complexesl’, in Molk, J. (red.), ‘Encyclopédie des Sciences Mathématiquesl’, Tome I, vol. 1, Fasc. 4, art. I5, pp. 329–468.
Chisholm, J.S.R. & Farwell, R.S. (1992), ‘Tetrahedral structure of idempotents of the Clifford algebra C3,1l’, in Micali A. et al. (eds.), ‘Clifford Algebras and their Applications in Mathematical Physicsl’, Proceedings of Second Workshop, Montpellier, France, 1989. Kluwer, Dordrecht, pp. 27–32.
Chisholm, J.S.R. & Farwell, R.S. (1993), ‘Spin gauge theories: principles and predictionsl’, in Brackx, F. et al. (eds.), ‘Clifford Algebras and their Applications in Mathematical Physicsl’, Proceedings of Third Conference, Deinze, Belgium, 1993. Kluwer, Dordrecht, pp. 367–374.
Conwell G.M. (1910), ‘The 3-space PG(3, 2) and its groupl’. Annals of Math. 11, pp. 60–76.
Coxeter, H.S.M. (1950), ‘Self-dual configurations and regular graphsl’. Bull. Amer. Math. Soc. 56, pp. 413–455.
Coxeter, H.S.M. (1958), ‘Twelve points in PG(5, 3) with 95040 self-transformationsl’. Proc. Roy. Soc. A. 247, pp. 279–293.
Crumeyrolle, A. (1990), ‘Orthogonal and Symplectic Clifford Algebras, Spinor Structuresl’. Kiuwer, Dordrecht.
Dickson, L.E. (1908), ‘Representations of the General Symmetric Group as Linear Groups in Finite and Infinite Fieldsl’. Trans. Amer. Math. Soc. 9, pp. 121–148.
Dye, R.H. (1992), ‘Maximal sets of non-polar points of quadrics and symplectic polarities over GF(2)l’. Geom. Ded. 44, pp. 281–293.
Eckmann, B. (1942), ‘Gruppentheoretischer Beweis des Satzes von Hurwitz-Radon über die Komposition quadratishcher Formenl’. Comment. Math. Helv. 15, pp. 358–366.
Eddington, A.S. (1936), ‘Relativity Theory of Protons and Electrons.’ Cambridge University Press, Cambridge.
Frenkel, I., Lepowsky, J. & Meurman A. (1988), ‘Vertex Operator Algebras and the Monsterl’. Academic Press, San Diego.
Gordon, N.A., Jarvis, T.M., Maks, J.G. & Shaw, R. (1991), ‘Composition algebras and GF’ (2Y. Mathematics Research Reports (University of Hull), Vol. IV, No. 19. (To be published in J. of Geometry.)
Griess, R.L. (1973), ‘Automorphisms of extra special groups and non vanishing degree 2 cohomologyl’. Pacific. J. Math. 48, pp. 403–422.
Griess, R.L. (1976), ‘On a subgroup of order 215. I GL(5, 2) 1 in E8(ℂ), the Dempwolff group and Aut(D8 o Dg o D8 )’. J. of Algebra 40, pp. 271–279.
Hagmark, P.-E. (1980), ‘Construction of some 2n-dimensional algebrasl’. Helsinki University of Technology, Mathematical Report HTKK-MAT-A177.
Hagmark, P.-E. & Lounesto, P. (1986), ‘Walsh functions, Clifford algebras and Cayley-Dickson processl’, in Chisholm, J.S.R. & Common, A.K. (eds), ‘Clifford Algebras and their Applications in Mathematical Physicsl’, Proceedings of First Workshop, Canterbury, England, 1985. Reidel, Dordrecht, pp. 531–540.
Hall, M (1976), ‘The Theory of Groupsl’. 2nd edn.. Chelsea, New York.
Hirschfeld, J.W.P. (1985), ‘Finite Projective Spaces of Three Dimensionsl’. Clarendon, Oxford.
Hirschfeld, J.W.P. & Thas, J.A. (1991), ‘General Galois Geometriesl’. Clarendon, Oxford.
Jacobson, N. (1980), ‘Basic Algebra IIl’. W.H.Freeman, San Francisco.
Janusz, G. & Rotman, J. (1982), ‘Outer automorphisms of S6’. Amer. Math. Monthly 89, pp. 407–410.
Lounesto, P. & Wene, G.P. (1987), ‘Idempotent structure of Clifford algebrasl’. Acta Applic. Math. 9, pp. 165–173.
Porteous, I.R. (1981), ‘Topological Geometryl’. Cambridge University Press, Cambridge.
Quillen, D. (1971), ‘The mod 2 cohomology ring of extra special 2-groups and the spinor groupsl’. Math Ann. 194, pp. 197–212.
Rotman, J. (1988), Amer. Math. Monthly 95, pp. 779–780, (solution to Advanced Problem 6538).
Shaw, R. (1983), ‘Linear Algebra and Group Representationsl’, Vol. 2, Academic Press, London.
Shaw, R. (1986), ‘The ten classical types of group representationsl’. J. Phys. A: Math. Gen. 19, pp. 35–44.
Shaw, R. (1987), ‘Ternary vector cross productsl’. J.Phys. A: Math. Gen. 20, pp. L689–L694.
Shaw, R. (1988), ‘A new view of d = 7 Clifford algebral’. J.Phys. A: Math Gen. 21, pp. 7–16.
Shaw, R. (1990), ‘Ternary composition algebras I, IIl’. Proc. R. Soc. Lond. A 431, pp. 1–19, pp. 21–36. ,
Shaw, R. (1992), ‘Finite geometries and Clifford algebras IIIl’, in Micali A. et al. (eds.), ‘Cliftord Algebras and their Applications in Mathematical Physicsl’, Proceedings of Second Workshop, Montpellier, France, 1989. Kluwer, Dordrecht, pp. 121–132.
Shaw, R. (1993), ‘Composition algebras, PG(m, 2) and non-split group extensionsl’, in del Ulmo, M.A. et al. (eds.), ‘Group Theoretical Methods in Physicsl’, Proceedings of the XIX International Colloquium, Salamanca, Spain, 1992, Vol. 1, CIEMAT, Madrid, pp. 467–470.
Shaw, R. (1994), ‘Finite geometries, Dirac groups and the table of real Clifford algebrasl’. Mathematics Research Reports (University of Hull), Vol. VII, No. 1.
Sylvester, J.J. (1844), ‘Elementary researches in the analysis of combinatorial aggregationl’. Philosophical Magazine XXIV, pp. 285–296.
Sylvester, J.J. (1861), ‘Notes on the historical origin of the unsymmetric six-valued function of six lettersl’. Philosophical Magazine XXI, pp. 369–377.
Tits, J. (1991), ‘Symmetrie’ in Hilton, P.J. et al. (eds.) ‘Miscellanea Mathematical’. Springer, Berlin, pp. 293–304.
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Shaw, R. (1995). Finite Geometry, Dirac Groups and the Table of Real Clifford Algebras. In: Ablamowicz, R., Lounesto, P. (eds) Clifford Algebras and Spinor Structures. Mathematics and Its Applications, vol 321. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8422-7_4
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DOI: https://doi.org/10.1007/978-94-015-8422-7_4
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