Orbits of the hyperoctahedral group as Euclidean designs
Béla Bajnok
Gettysburg College, Gettysburg, Pennsylvania USA
DOI: 10.1007/s10801-006-0042-3
Abstract
The hyperoctahedral group H in n dimensions (the Weyl group of Lie type B n ) is the subgroup of the orthogonal group generated by all transpositions of coordinates and reflections with respect to coordinate hyperplanes.With e 1 , ..., e n denoting the standard basis vectors of \sf R \sf{R} n and letting x k = e 1 + \cdot \cdot \cdot + e k ( k = 1, 2, ..., n), the set
| $ {\cal I}^n_k={\bf x}_{\bf k}^H=\{ {\bf x}_{\bf k}^g \mbox{} | \mbox{} g \in H \}$ {\cal I}^n_k={\bf x}_{\bf k}^H=\{ {\bf x}_{\bf k}^g \mbox{} | \mbox{} g \in H \}
Pages: 375–397 Keywords: keywords Euclidean design; spherical design; tight design; harmonic polynomial; hyperoctahedral group Full Text: PDF References1. B. Bajnok, “Construction of spherical t-designs,” Geom. Dedicata 43/2 (1992), 167-179.
2. B. Bajnok, “On Euclidean designs,” Adv. Geom. 6 (2006) 431-446. 3. Ei. Bannai, “On extremal finite sets in the sphere and other metric spaces,” London Math. Soc. Lecture Note Ser. 131 (1988) 13-38. 4. Ei. Bannai and Et. Bannai, “On Euclidean tight 4-designs,” Preprint. 5. Et. Bannai, “On antipodal Euclidean tight (2e + 1)-designs,” Preprint. 6. J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed. Springer-Verlag, 1999. 7. P. de la Harpe and C. Pache, “Cubature formulas, geometrical designs, reproducing kernels, and Markov operators,” Preprint. 8. P. de la Harpe, C. Pache, and B. Venkov, “Construction of spherical cubature formulas using lattices,” Preprint. 9. P. Delsarte, J.M. Goethals, and J.J. Seidel, “Spherical codes and designs,” Geom. Dedicata 6/3 (1977) 363-388. 10. P. Delsarte and J.J. Seidel, “Fisher type inequalities for Euclidean t-designs,” Linear Algebra Appl. 114-115 (1989), 213-230. 11. A. Erdélyi et al., Higher Transcendental Functions, (Bateman Manuscript Project), MacGraw-Hill, Vol. 2, 1953. 12. T. Ericson and V. Zinoviev, “Codes on Euclidean spheres,” Elsevier, 2001. 13. C.D. Godsil, Algebraic Combinatorics, Chapman and Hall, Inc., 1993. 14. J.M. Goethals and J.J. Seidel, “Spherical designs,” in Relations Between Combinatorics and Other Parts of Mathematics, Proc. Sympos. Pure Math., D.K. Ray-Chaudhuri (Ed.), Amer. Math. Soc., Providence, RI, 1979, vol. 34, pp. 255-272. 15. J.M. Goethals and J.J. Seidel, “Cubature formulae, polytopes and spherical designs,” in The Geometric Vein: The Coxeter Festschrift, C. Davis, B. Gr\ddot unbaum, and F.A. Sher (Eds.), Springer-Verlag, New York-Berlin, 1981, pp. 203-218. 16. F.B. Hildebrand, Introduction to Numerical Analysis, 2nd ed. McGraw-Hill, Inc., 1974. 17. A. Neumaier and J.J. Seidel, “Discrete measures for spherical designs, eutactic stars and lattices,” Nederl. Akad. Wetensch. Indag. Math. 50/3 (1988), 321-334. 18. J.J. Seidel, “Designs and approximation,” Contemp. Math. 111 (1990), 179-186. 19. J.J. Seidel, “Isometric embeddings and geometric designs,” Discrete Math. 136 (1994), 281-293. 20. J.J. Seidel, “Spherical designs and tensors,” in Progress in algebraic combinatorics, Adv. Stud. Pure Math., E. Bannai and A. Munemasa (Eds.), Math. Soc. Japan, Tokyo, 1996, vol. 24, pp.309-321. 21. P.D. Seymour and T. Zaslavsky, “Averaging sets: A generalization of mean values and spherical designs,” Adv. Math. 52 (1984), 213-240. 22. A.H. Stroud, Approximate Calculation of Multiple Integrals, Prentice-Hall, Inc., 1971.
© 1992–2009 Journal of Algebraic Combinatorics
|