Designs Related Through Projective and Hopf Maps

We verify a construction which, for K K the reals, complex numbers, quaternions, or octonions, builds a spherical t -design by placing a spherical t -design on each K K -projective or K K -Hopf fiber associated to the points of a t/2 ⌊ t / 2 ⌋ -design on a quotient projective space KPⁿ≠ OP² K P n ≠ O P 2 or sphere. This generalizes work of König and Kuperberg, who verified the K= C 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 K = C case of the generalized Hopf settings.