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Coxeter-Todd Lattice

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The Coxeter-Todd lattice is the complex lattice Lambda_6^omega corresponding to the real lattice K_(12), where omega=e^(2pii/3). It gives the densest known hypersphere packing in twelve dimensions (Conway and Sloane 1983, Cohn). Its kissing number is 756 (Nebe and Sloane). The associated automorphism group G_0 was discovered by Mitchell (1914). The group order of G_0 is given by

|Aut(Lambda_6^omega)|=2^9·3^7·5·7=39191040.
(1)

The group order of the automorphism group of K_(12) is given by

|Aut(K_(12))|=2^(10)·3^7·5·7
(2)

(Conway and Sloane 1993).

The theta series for the Coxeter-Todd lattice is given in terms of Jacobi theta functions by

f(q)=9/(32)theta_2^6(q)theta_2^6(q^3)+[theta_2(q^4)theta_2(q^(12))+theta_3(q^4)theta_3(q^(12))]^6+(45)/(16)theta_2^4(q)[theta_2(q^4)theta_2(q^(12))+theta_3(q^4)theta_3(q^(12))]^2theta_2^4(q^3) =1+756q^4+4032q^6+20412q^8+60480q^(10)+...
(3)

(OEIS A004010), where |q|<1.

Properties of the Coxeter-Todd lattice are implemented in the Wolfram Language as LatticeData["CoxeterTodd", prop].

See also

Barnes-Wall Lattice, Leech Lattice, Modular Euclidean Lattice, Theta Series

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References

Cohn, H. "Sphere Packing." https://cohn.mit.edu/sphere-packing/.Conway, J. H. and Sloane, N. J. A. "The Coxeter-Todd Lattice, the Mitchell Group and Related Sphere Packings." Math. Proc. Camb. Phil. Soc. 93, 421-440, 1983. https://doi.org/10.1017/S0305004100060746.Conway, J. H. and Sloane, N. J. A. "The 12-Dimensional Coxeter-Todd Lattice K_(12)." §4.9 in Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, pp. 127-129, 1993.Coxeter, H. S. M. and Todd, J. A. "An Extreme Duodenary Form." Canad. J. Math. 5, 384-392, 1953. https://doi.org/10.4153/CJM-1953-043-4.Mitchell, H. H. "Determination of All Primitive Collineation Groups in More than Four Variables." Amer. J. Math. 36, 1-12, 1914.Nebe, G. and Sloane, N. J. A. "The Lattice K12 (The Coxeter-Todd Lattice)." https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/K12.html.Sloane, N. J. A. Sequence A004010/M5478 in "The On-Line Encyclopedia of Integer Sequences."Todd, J. A. "The Characters of a Collineation Group in Five Dimensions." Proc. Roy. Soc. London Ser. A 200, 320-336, 1950.

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Coxeter-Todd Lattice

Cite this as:

Weisstein, Eric W. "Coxeter-Todd Lattice." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Coxeter-ToddLattice.html

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