Designs Related Through Projective and Hopf Maps

Abstract We verify a construction which, for $$\mathbb K$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> the reals, complex numbers, quaternions, or octonions, builds a spherical t -design by placing a spherical t -design on each $$\mathbb K$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> -projective or $$\mathbb K$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> -Hopf fiber associated to the points of a $$\lfloor t/2\rfloor $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⌊</mml:mo> <mml:mi>t</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>⌋</mml:mo> </mml:mrow> </mml:math> -design on a quotient projective space $$\mathbb{K}\mathbb{P}^n\ne \mathbb{O}\mathbb{P}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>≠</mml:mo> <mml:mi>O</mml:mi> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> or sphere. This generalizes work of König and Kuperberg, who verified the $$\mathbb K=\mathbb C$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>=</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> case of the projective settings, and of Okuda, who (inspired by independent observation of this construction by Cohn, Conway, Elkies, and Kumar) verified the $$\mathbb K=\mathbb C$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>=</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> case of the generalized Hopf settings.