Harmonic optima

This is a list of some of the most noteworthy point configurations from the table of records for harmonic energy. Each row describes a configuration of N points on the unit sphere in n dimensions; in the list of inner products, an ordered pair (t, m) means the inner product t occurs m times on average around each point. In other words, if for each point x in the configuration we count the number of y that have inner product t with x, then the average over all x is m. This information suffices to compute the energy under any radial pair potential.

Universal optima

These configurations are known to minimize energy for all potential functions that are completely monotonic functions of squared Euclidean distance, by Theorem 1.2 in Universally optimal distribution of points on spheres by Henry Cohn and Abhinav Kumar (J. Amer. Math. Soc. 20 (2007), 99-148). In particular, they minimize harmonic energy, or more generally energy under any inverse power law. Aside from regular polygons, simplices, and cross polytopes, no other universal optima are known to exist in less than 52 dimensions (see Table 1 in that paper).

N n Inner products
12 3 (-1, 1), (-1/51/2, 5), (1/51/2, 5)
120 4 (-1, 1), ((-1-51/2)/4, 12), (-1/2, 20), ((1-51/2)/4, 12), (0, 30), ((-1+51/2)/4, 12), (1/2, 20), ((1+51/2)/4, 12)
16 5 (-3/5, 5), (1/5, 10)
27 6 (-1/2, 10), (1/4, 16)
56 7 (-1, 1), (-1/3, 27), (1/3, 27)
240 8 (-1, 1), (-1/2, 56), (0, 126), (1/2, 56)
112 21 (-1/3, 30), (1/9, 81)
100 22 (-4/11, 22), (1/11, 77)
275 22 (-1/4, 112), (1/6, 162)
891 22 (-1/2, 42), (-1/8, 512), (1/4, 336)
552 23 (-1, 1), (-1/5, 275), (1/5, 275)
4600 23 (-1, 1), (-1/3, 891), (0, 2816), (1/3, 891)
196560 24 (-1, 1), (-1/2, 4600), (-1/4, 47104), (0, 93150), (1/4, 47104), (1/2, 4600)

Conjectural optima for harmonic energy

These point configurations appear to minimize harmonic energy. Please let me know if you can improve on any of them. Among this list, the configurations of 40 points in 10 dimensions and 64 points in 14 dimensions are especially intriguing, since they appear to be universally optimal, as discussed in the paper on this data set.

N n Inner products
32 3 (-1, 1), (-(75+30⋅51/2)1/2/15, 15/4), (-51/2/3, 15/8), (-1/51/2, 15/8), (-1/3, 15/4), (-(75-30⋅51/2)1/2/15, 15/4), ((75-30⋅51/2)1/2/15, 15/4), (1/3, 15/4), (1/51/2, 15/8), (51/2/3, 15/8), ((75+30⋅51/2)1/2/15, 15/4)
6 4 (-1/2, 2), (0, 3)
6 4 (-1, 1/3), (-1/3, 2), (0, 8/3)
10 4 (-2/3, 3), (1/6, 6)
10 4 ((-1-51/2)/4, 2), (0, 5), ((-1+51/2)/4, 2)
13 4 ((cos(8π/13)+cos(12π/13))/2, 4), ((cos(2π/13)+cos(10π/13))/2, 4), ((cos(4π/13)+cos(6π/13))/2, 4)
15 4 (-1/21/2, 12/5), (-1/2, 16/5), (0, 6/5), (1/4, 12/5), (1/81/2, 24/5)
24 4 (-1, 1), (-1/2, 8), (0, 6), (1/2, 8)
48 4 (-1, 1), (-1/21/2, 2), ((-1-31/2)/4, 4), (-(3/8)1/2, 4), (-1/81/2, 4), ((1-31/2)/4, 4), (0, 10), ((-1+31/2)/4, 4), (1/81/2, 4), ((3/8)1/2, 4), ((1+31/2)/4, 4), (1/21/2, 2)
21 5 (-(2/5)1/2, 20/7), (-1/2, 30/7), (-1/5, 10/7), (1/4, 40/7), (1/101/2, 40/7)
32 5 (-1, 1), (-1/51/2, 15/2), (-1/3, 45/8), (-1/5, 15/8), (1/5, 15/8), (1/3, 45/8), (1/51/2, 15/2)
42 6 (-1, 1), (-2/5, 10), (-3/10, 10), (3/10, 10), (2/5, 10)
44 6 (-1, 1), (-1/61/2, 96/11), (-1/3, 120/11), (0, 30/11), (1/3, 120/11), (1/61/2, 96/11)
126 6 (-1, 1), (-(3/8)1/2, 96/7), (-1/2, 110/7), (-1/4, 48/7), (0, 360/7), (1/4, 48/7), (1/2, 110/7), ((3/8)1/2, 96/7)
78 7 (-1, 7/39), (-5/7, 224/39), (-1/71/2, 448/39), (-1/7, 1120/39), (0, 28/13), (1/71/2, 448/39), (3/7, 224/13)
148 7 (-1, 21/37), (-5/7, 112/37), (-(2/7)1/2, 672/37), (-1/2, 420/37), (-1/7, 560/37), (0, 2226/37), (3/7, 336/37), (1/2, 420/37), ((2/7)1/2, 672/37)
182 7 (-1, 1), (-1/31/2, 216/13), (-1/2, 288/13), (-1/3, 108/13), (0, 1116/13), (1/3, 108/13), (1/2, 288/13), (1/31/2, 216/13)
72 8 (-1, 1), (-5/14, 14), (-2/7, 21), (2/7, 21), (5/14, 14)
96 9 (-1, 1), (-1/3, 39), (0, 16), (1/3, 39)
40 10 (-1/2, 8), (-1/3, 3), (0, 4), (1/6, 24)
42 14 (-1/2, 6), (-1/5, 5), (1/10, 30)
64 14 (-3/7, 14), (-1/7, 7), (1/7, 42)
112 15 (-1/3, 36), (-1/15, 15), (1/5, 60)
128 15 (-1/3, 42), (-1/15, 15), (1/5, 70)
256 16 (-1, 1), (-1/4, 112), (0, 30), (1/4, 112)
88 18 (-1/61/2, 96/11), (-1/3, 153/11), (0, 60/11), (1/9, 360/11), (1/541/2, 288/11)
56 20 (-2/5, 10), (1/15, 45)
120 20 (-2/7, 42), (1/7, 77)
77 21 (-3/8, 16), (1/12, 60)
162 21 (-2/7, 56), (1/7, 105)
336 21 (-3/5, 5), (-1/5, 160), (1/5, 170)