We verify a construction which, for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb K$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">K</mi> </math></EquationSource> </InlineEquation> the reals, complex numbers, quaternions, or octonions, builds a spherical t-design by placing a spherical t-design on each <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb K$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">K</mi> </math></EquationSource> </InlineEquation>-projective or <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb K$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">K</mi> </math></EquationSource> </InlineEquation>-Hopf fiber associated to the points of a <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\lfloor t/2\rfloor $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>⌊</mo> <mi>t</mi> <mo stretchy="false">/</mo> <mn>2</mn> <mo>⌋</mo> </mrow> </math></EquationSource> </InlineEquation>-design on a quotient projective space <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb{K}\mathbb{P}^n\ne \mathbb{O}\mathbb{P}^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">K</mi> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mi>n</mi> </msup> <mo>≠</mo> <mi mathvariant="double-struck">O</mi> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> or sphere. This generalizes work of König and Kuperberg, who verified the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb K=\mathbb C$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">K</mi> <mo>=</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation> case of the projective settings, and of Okuda, who (inspired by independent observation of this construction by Cohn, Conway, Elkies, and Kumar) verified the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb K=\mathbb C$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">K</mi> <mo>=</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation> case of the generalized Hopf settings.

Designs Related Through Projective and Hopf Maps

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2025年11月28日

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Ayodeji Lindblad

摘要

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