Papers › Constructing high order spherical designs as a union of two of lower order
Constructing high order spherical designs as a union of two of lower order
Mozhgan Mohammadpour, Shayne Waldron
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We show how the variational characterisation of spherical designs can be used to take a union of spherical designs to obtain a spherical design of higher order (degree, precision, exactness) with a small number of points. The examples that we consider involve taking the orbits of two vectors under the action of a complex reflection group to obtain a weighted spherical (t,t)-design. These designs have a high degree of symmetry (compared to the number of points), and many are the first known construction of such a design, e.g., a $32$ point (9,9)-design for ℂ², a $48$ point (4,4)-design for ℂ³, and a $400$ point (5,5)-design for ℂ⁴.From a real reflection group, we construct a $360$ point (9,9)-design for ℝ⁴ (spherical half-design of order $18$), i.e., a $720$ point spherical $19$-design for ℝ⁴.
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