On spherical designs, a survey Eiichi Bannai Shanghai Jiao Tong University 2016 April 21 at SJTU The same talk was given at Kinki University, Feb 15, 2016 Osaka, Japan 1
Main References 1. P. Delsarte, J.M. Goethals and J. J. Seidel: Spherical codes and designs, Geom. Dedicata 6 (1977), 363–388. 2. 坂内英一・坂内悦子、球面上の代数的組合せ理論、シュプリンガー 東京、１９９９ . 3. Eiichi Bannai and Etsuko Bannail: A survey on spherical designs and algebraic combinatorics on spheres, Europ. J. Combinatorics, 30 (2009), 1392–1425. 4. H. Cohn and A. Kumar: Universally optimal distribution of points on spheres, J. Amer. Math. Soc. 20 (2007), 99–148. 5. H. Cohn: Order and disorder in energy minimization, Proceedings of ICM, Hyderbad, India, 2010, Volume IV, pages 2416–2443. 2
S n − 1 = { ( x 1 , x 2 , · · · , x n ) ∈ R n | x 2 1 + x 2 2 + · · · + x 2 n = 1 } . X ⊂ S n − 1 , | X | < ∞ . What X are good? Answer may not be unique. There are several different viewpoints (or criterions) about this question. What criterions are ”good” is a part of the question. 3
Definition (Optimal code). X ( ⊂ S n − 1 ) is called an optimal code if the minimum distance Min { d ( x, y ) | x, y ∈ X, x ̸ = y } is max- imum among all the subsets (on S n − 1 ) of the same cardinality. Optimal codes do exist for each given pair of n and | X | . (They may not be unique.) Classifications of optimal codes on S 2 are known for | X | ≤ 12 and | X | = 24 . (See the book, Ericson-Zinoviev: Codes on Euclidean Spheres, 2001.) Optimal codes on S n − 1 , n ≥ 4 are deter- mined only for very special cases. (Many candidates are given by computer simulations.) 4
Recently, Musin-Tarazov solved the case of n = 3 and | X | = 13 . (Strong thirteen spheres problem. Discrete & Computational Geome- try, 48:1 (2012), 128-141) More recently, Musin-Tarazov announced the solution of the case of n = 3 and | X | = 14 . (The Tammes problem for N = 14 , arXiv: 1410:2536.) Finding optimal codes is also called Tammes’ problem, named after the Dutch botanist who posed the problem in connection with the study of pores in spherical pollen grains (1930). 5
Find X ⊂ S n − 1 with the Coulomb-Thomson problem. property that 1 ∑ || x − y || x,y ∈ X,x ̸ = y is minimum among all the subsets on S n − 1 of the same cardinality as | X | , where ( x 1 − y 1 ) 2 + ( x 2 − y 2 ) 2 + · · · + ( x n − y n ) 2 . √ || x − y || = We can also consider energy minimizing sets X on S n − 1 for other potential functions, for example, 1 ∑ || x − y || k , ( k ∈ Z > 0 ) x,y ∈ X,x ̸ = y is minimum among all the subsets on S n − 1 of the same cardinality as | X | . 6
Energy minimizing subset (w.r.t. f). Let f : (0 , 4] − → R ≥ 0 be a decreasing function. Then X is called an energy minimizing set (on S n − 1 ) w.r.t. f , if ∑ f ( || x − y || 2 ) x,y ∈ X,x ̸ = y is minimum , among all the subsets on S n − 1 of the same cardinality as | X | . It is Coulomb-Thomson problem, if we take f ( r ) = r − 1 2 . (Of course, for different functions f and g , X may be an energy min- imizing set for f , but not an energy minimizing set for g. ) Are there X ⊂ S n − 1 which are energy minimizing sets for all reasonably good classes of functions f : (0 , 4] − → R ≥ 0 ? 7
Definition (Universally optimal codes) . (Cohn-Kumar, 2007). X ⊂ S n − 1 is called a universally optimal code (on S n − 1 ), if X is an energy minimizing set, among all the subsets (on S n − 1 ) of the same cardinality as X , w.r.t. all completely monotonic (decreasing) functions f : (0 , 4] − → R ≥ 0 , where f is called completely monotonic, if f ∈ C ∞ and ( − 1) k f ( k ) ( r ) ≥ 0 for all k = 0 , 1 , 2 , · · · . That is, ∑ ∑ f ( || x − y || 2 ) ≤ f ( || x − y || 2 ) x,y ∈ X,x ̸ = y x,y ∈ Y,x ̸ = y for all Y ⊂ S n − 1 with | Y | = | X | and for all complete monotonic function f : (0 , 4] − → R ≥ 0 . 8
Another equivalent definition of universally opti- mal codes. X ⊂ S n − 1 becomes a universally optimal code (on S n − 1 ), if X satisfy ∑ ∑ f ( x · y ) ≤ f ( x · y ) x,y ∈ X,x ̸ = y x,y ∈ Y,x ̸ = y for all Y ⊂ S n − 1 with | Y | = | X | and for all absolutely monotonic (increasing) function f : [ − 1 , 1) − → R ≥ 0 . Here x · y is the usual Euclidean inner product, and f is called absolutely monotonic , if f ∈ C ∞ and f ( k ) ( t ) ≥ 0 for all k = 0 , 1 , 2 , · · · . 9
Remarks and Examples. (i) f ( r ) = r − s , s > 0 are complete monotonic functions. f ( r ) = (4 − r ) k , k = 1 , 2 , . . . are complete monotonic functions. (ii) Universally optimal codes are optimal codes. (Consider the function f ( r ) = r − s , with s → ∞ . ) (iii) For n = 2, i.e., for S 1 ⊂ R 2 , the set of vertices of a regular polygon is a universally optimal code. (These are the only universally optimal codes on S 1 . (iv) For n = 3, the sets of vertices of regular tetrahedron, octahedron, and icosahedron, are universally optimal codes, (4, 6, 12 vertices, re- spectively.) Together with the trivial cases of up to 3 points, they are the only universally optimal codes on S 2 . Also, note that the sets of vertices of cube and dodecahedron (8, 20 vertices, respectively) are not universally optimal. They are not even optimal. 10
(v) The set of 240 roots of type E 8 (240 min. vectors of E 8 -lattice) in R 8 gives a universally optimal code. (vi) The set of 196560 min. vectors of Leech lattice in R 24 gives a universally optimal code. (vii) There are some more examples of universally optimal codes, but they are fairly rare. Cohn-Kumar (2007) conjectures that there are only finitely many universally optimal codes for each n ≥ 4 . (The classifications are open for any dimension n ≥ 4.) The next table gives known universally optimal codes (after Levenshtein and Cohn-Kumar) 11
n | X | t tight inner product Name 2 N N − 1 yes cos(2 πj/N ) (1 ≤ j ≤ N/ 2) regular N-gon n N ≤ n 1 no − 1 / ( N − 1) regular simplex n n + 1 2 yes − 1 /n regular simplex n 2 n 3 yes − 1 , 0 regular cross polytope √ 3 12 5 yes − 1 , ± 1 / 5 regular icosahedron √ 4 120 11 no − 1 , ± 1 / 2 , 0 , ( ± 1 ± 5) / 4 regular 600-cell 5 16 3 no − 3 / 5 , 1 / 5 Clebsch graph(=hemicube) 6 27 4 yes − 1 / 2 , 1 / 4 Schl¨ afli graph 7 56 5 yes − 1 , ± 1 / 3 28 equangular lines 8 240 7 yes − 1 , ± 1 / 2 , 0 E 8 root system 21 112 3 no − 1 / 3 , 1 / 9 U 4 (3) isotropic subspaces 21 162 3 no − 2 / 7 , 1 / 7 U 4 (3) /L 4 (3) , strong regular graph 22 100 3 no − 4 / 11 , 1 / 11 Higman-Sims graph 22 275 4 yes − 1 / 4 , 1 / 6 MacLaughlin graph 22 891 5 no − 1 / 2 , − 1 / 8 , 1 / 4 generalized hexagon 23 552 5 yes − 1 , ± 1 / 5 276 equangular lines 23 4600 7 yes − 1 , ± 1 / 3 , 0 iterated kissing configuration 24 196560 11 yes − 1 , ± 1 / 2 , ± 1 / 4 , 0 min. vectors of Leech lattice ( q + 1)( q 3 + 1) q q 3 +1 3 ∗ no ∗ − 1 /q, 1 /q 2 U 3 ( q ) isotropic subspaces q +1 ( q is a prime power. The cases q = 2 ( t = 4 and tight) and q = 3 are already in the list.) 12
Definition (Spherical t -designs). Delsarte-Goethals-Seidel (1977) Let t be a positive integer. X ⊂ S n − 1 , | X | < ∞ , is called a spherical t -design on S n − 1 , if the following condition is satisfied. 1 1 ∫ ∑ S n − 1 f ( x ) dσ ( x ) = f ( x ) | S n − 1 | | X | x ∈ X for any polynomials f ( x ) = f ( x 1 , x 2 , . . . , x n ) of degrees up to t . Here, | S n − 1 | denotes the area of the sphere S n − 1 , and the integral is the surface integral on S n − 1 . We are interested in the t -designs with the cardi- nality | X | as small as possible. 13
Theorem (Delsarte-Goethals-Seidel, 1977). ( n − 1+ e ( n − 1+ e − 1 ) ) X = 2 e -design = ⇒ | X | ≥ + . e e − 1 ( n − 1+ e ) X = (2 e + 1)-design = ⇒ | X | ≥ 2 . e (These inequalities are called Fisher type inequalities .) X is called a tight t-design if equality ” = ” holds in one of the above two inequalities. 14
Classifications of tight t -designs. (i) Tight t -designs on S 1 are regular ( t + 1)-gons. (ii) If a tight t -design on S n − 1 exists for n ≥ 3 , then t ∈ { 1 , 2 , 3 , 4 , 5 , 7 , 11 } (Bannai-Damerell (1979/80), Bannai-Sloane (1981). (iii) tight t -designs are completely classified for t ≤ 3 and t = 11 . (iv) There are some developments of the classification of tight t -designs for t = 4 , 5 , 7 . See Bannai-Munemasa-Venkov, The nonexistence of certain tight spher- ical designs, Algebra i Analiz(2004). See also, G. Nebe and B. Venkov, On tight spherical designs, Algebra i Analiz (2012). The complete classifications are still open for t = 4 , 5 , 7 . 15
Remarks . (i) Let X ⊂ S n − 1 . X is called an s -distance set if |{ d ( x, y ) | x, y ∈ X, x ̸ = y }| = s. (ii) Suppose X is an s -distance set and t -design, then we have t ≤ 2 s. Moreover, we have t = 2 s ⇐ ⇒ X = tight 2 s -design . and X is antipodal, and t = 2 s − 1 ⇐ ⇒ X = tight (2 s − 1) -design . 16
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