2019/12/16 by Mozhgan Mohammadpour, Mohammadpour, Mozhgan, Shayne Waldron +1
Mathematics · #05B30 #42C15 #65D30 #94A12 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #math.CO #math.MG #msc:05B30 #msc:42C15 #msc:65D30 #msc:94A12
paper · pdf · doi:10.48550/arxiv.1912.07151
21 pages
arxiv created 2019/12/16 · arxiv updated 2019/12/17
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.