Spherical 7-Designs in 2 n-Dimensional Euclidean Space
V.M. Sidelnikov
DOI: 10.1023/A:1018723416627
Abstract
We consider a finite subgroup 
n of the group O(N) of orthogonal matrices, where N = 2 n , n = 1, 2 .... This group was defined in [7]. We use it in this paper to construct spherical designs in 2 n -dimensional Euclidean space R N . We prove that representations of the group n on spaces of harmonic polynomials of degrees 1, 2 and 3 are irreducible. This and the earlier results [1-3] imply that the orbit n,2 x t of any initial point x on the sphere S N - 1 is a 7-design in the Euclidean space of dimension 2 n .
n of the group O(N) of orthogonal matrices, where N = 2 n , n = 1, 2 .... This group was defined in [7]. We use it in this paper to construct spherical designs in 2 n -dimensional Euclidean space R N . We prove that representations of the group n on spaces of harmonic polynomials of degrees 1, 2 and 3 are irreducible. This and the earlier results [1-3] imply that the orbit n,2 x t of any initial point x on the sphere S N - 1 is a 7-design in the Euclidean space of dimension 2 n .Pages: 279–288
Keywords: spherical design; orthogonal matrix; euclidian space
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References
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2. J.M. Goethals and J.J. Seidel, “Spherical designs,” Proc. of Symposium in Pure Math., Vol. 34, pp. 255-272, 1979.
3. E. Bannai, “On some spherical t-designs,” J. Combinatorial Theory (A) 26 (1979), 157-161.
4. E. Bannai, “Spherical t-designs which are orbits of finite groups,” J. Math. Soc. 36(2), 1984, 341-354.
5. S. Helgason, Groups and Geometrical Analysis, Academic Press, INC, 1984.
6. N. Ya. Vilenkin, Special Functions and Theory of Representation of Groups, Moscow, Nauka, 1965, (in Russian).
7. V.M. Sidelnikov, “On one finite matrix group and codes on Euclidean sphere,” Problems of Information Transmission 33 (1997), 29-44.
8. J.H. Conway, and N.J.A. Sloane, Sphere Packing, Lattices and Groups, 2nd edition, Springer-Verlag, 1993.
9. Lev Kazarin, “On groups suggested by Sidelnikov,” Math. USSR Sbornik (1998), to appear.