Designs Related Through Projective and Hopf Maps

Abstract

We verify a construction which, for K the reals, complex numbers, quaternions, or octonions, builds a spherical t-design by placing a spherical t-design on each K -projective or K -Hopf fiber associated to the points of a ⌊ t / 2 ⌋ -design on a quotient projective space K P n ≠ O P 2 or sphere. This generalizes work of König and Kuperberg, who verified the K = C case of the projective settings, and of Okuda, who (inspired by independent observation of this construction by Cohn, Conway, Elkies, and Kumar) verified the K = C case of the generalized Hopf settings.