Spherical designs and finite group representations (some results of E. Bannai)

Authors:
Pierre de la Harpe;Claude Pache
Affiliations:
Section de Mathématiques, Université de Genève, C.P. 240, CH-1211 Genève 24, Switzerland;Section de Mathématiques, Université de Genève, C.P. 240, CH-1211 Genève 24, Switzerland
Venue:
European Journal of Combinatorics - Special issue on algebraic combinatorics: in memory of J.J. Seidel
Year:
2004

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Abstract

We reprove several results of Bannai concerning spherical t-designs and finite subgrourps of orthogonal groups. These include criteria in terms of harmonic representations of subgroups of O(n) for the corresponding orbits to be t-designs (t = 0, 1, 2, 3, ...) in Sn - 1. We also discuss a conjecture of Bannai, dating from 1984, according to which t is bounded independently of the dimension n (for n ≥ 3) for such designs.