Spherical designs and some generalizations : An overview

Transcription

1 Spherical designs and some generalizations : An overview Eiichi Bannai Shanghai Jiao Tong University 2015 April 21 Workshop on spherical designs and numerical analysis Shanghai, China

2 Spherical t-designs 1 S n 1 = {(x 1, x 2,, x n ) R n x x x2 n = 1}. Y S n 1, 0 < Y <. Definition (Delsarte-Goethals-Seidel,1977) Y is called a spherical t-design on S n 1 if and only if 1 f(x)dσ(x) = 1 f(x) S n 1 S n 1 Y x Y for any polynomials f(x) = f(x 1, x 2,..., x n ) of degree t.

3 Some generalizations 2 (1). Replacing the sphere S n 1 with other spaces. (i) compact symmetric spaces of rank 1 (= Projective spaces over R, C, H, O) (Further generalization: Designs on Grasmannian spaces, cf. talk of Martin Ehler ) (ii) ( V k) = Johnson J(v, k) association scheme (iii) F n 2 = Hamming association scheme H(n, 2) (or Q-polynomial association schemes) (2). Replacing the space of Polynomials of degree at most t by other set of functions But I will not talk much about these in this talk.

4 3 weighted spherical t-designs on S n 1 (=cubature formula of degree t) (II) Spherical t-designs (I) allow several concentric spheres (=Euclidean t-designs) (III)

5 (II) weighted spherical t-designs in S n 1 (=cubature formula of degree t on S n 1 ) Y S n 1, 0 < Y < w : Y R >0 4 (Y, w) is a weighted spherical t-design on S n 1 1 f(x)dσ(x) = w(y)f(y), S n 1 S n 1 y Y for f(x) = f(x 1,..., x n ), polynomial of degree t

6 (III) allow several concentric spheres (=Euclidean t-designs) 5 Y S n 1 (r 1 ) S n 1 (r 2 ) S n 1 (r p ) 0 < Y < w : Y R >0 Definition: (Y, w) is a Euclidean t-design p w(y S n 1 (r i )) i=1 f(x)dσ(x) = S n 1 (r i ) S n 1 (r i ) y Y f(x) = polynomial of degree t w(y)f(y),

7 Fisher type inequalities (I), (II) Y = t-design or (Y, w)=weighted t-design (III) (Y, w) = Euclidean t-design = Y = Y if t = 2e, 6 ( n 1+e ) ( e + n 1+e 1 ) e 1 = me, with m e = m e + m e m 1 + m 0 m i = dim(harm i (R n )) = ( n 1+i) ( i n 1+i 2 ) i 2 if t = 2e ( n 1+e) e if t = 2e, m e + m e m e p+1 if t = 2e + 1 more complicated

8 Tight designs: Those satisfying one of the equality in above 7 (I) tight spherical t-designs Delsarte-Goethals-Seidel (1977) Bannai-Damerell (1979/1980) Bannai-Sloane (1981) Bannai-Munemasa-Venkov (2004) Nebe-Venkov (2012) still open for t = 4, 5, 7 (II) tight weighted spherical t-designs = tight spherical t-design (reduced to (I))

9 (III) Classification problems of tight Euclidean t-designs 8 Neumaier-Seidel (1988) Delsarte-Seidel (1989) Bajnok (2006) B-B ( ) B-B-Hirao-Sawa (2010) many papers B-B (2014) Moscow J. of Combinatorics and Number Theory mostly on two shells (spheres) (The results are still very partial!)

10 Another generalization 9 (IV) spherical designs of harmonic index t Y S n 1, 0 < Y <, f(y) = 0, for f(y) Harm t (R n ). y Y (Note that Y is a spherical t-design f(y) = 0, for f(y) Harm i (R n ) and 1 i t.) y Y

11 Fisher type in equality for (IV) 10 Q i (x) = Gegenbauer polynomial of degree i. i.e. 1 Q i (x)q j (x)(1 x 2 ) n 3 2 = α i δ i,j, α i > 0 1 Q i (1) = ( n 1+i) ( i n 1+i 2 ) i 2 = dim(harmi (R n ) c 2e = min. value of Q 2e (x) Q 0 (x) = 1 Q 1 (x) = nx Q 2 (x) = n+2 2 (nx2 1) Q 3 (x) = n(n+4) ((n + 2)x 2 3) 3! Q 4 (x) = n(n+6) ((n + 4)(n + 2)x 4 6(n + 2)x 2 + 3) 4!

12 (The graph of y = Q 8 (x)) 11

13 Y = a spherical designs of harmonic index t. Let A(n, t) = smallest Y such that Y is a spherical design of harmonic index t on S n (1) t = 2e + 1 = Y 2, since Y = {y, y} is a design of harmonic index 2e + 1 on S n 1 So, A(n, 2e + 1) = 2 (2) t = 2 = Y n, Y = {e 1, e 2,..., e n } is a design of harmonic index 2 on S n 1 (3) n = 2 (t = 2e) = Y 2, x, y S 1 with the angle θ = π/2e is such an example So, A(2, 2e) = 2 for e

14 First nontrivial case: A(3, 4) Let m i = 2i + 1, and let f 1, f 2,..., f 2i+1 be an orthonormal basis of Harm i (R 3 ). Our case is i = 4. Then minimize { 9 j=1( y Y f j (y)) 2 Y S 2, Y is fixed.} (Note that this quantity is = 0, if and only if Y is a design of harmonic index 4 on S 2.) Y = 2, 3, 4 are shown to be impossible. Y = 5. Examples were found by numerical experiments. (Namely, we obtained regular pentagons of two different sizes.) 13

15 Theorem (B-Okuda-Tagami) (to appear in J. Approx. Theory (2015)) Y = a spherical 2e -design on S n 2, r = a root of Q 2e (x), Let Y S n 2 be defined by 14 Y = {(r, 1 r 2 x) S n 1 x Y }. Then Y is a design of harmonic index 2e on S n 1 Theorem (B-O-T) A(3, 4) = 5. Moreover, if Y is a design of harmonic index 4 on S 2 ( R 3 ), and Y = 5. Then Y is a pentagon on the circle of radius 1 r 2, where r is a zero of Q 4 (x) (for n = 3), i.e. r 1, r 2 = or those with any points replaced by their antipodal points.

16 15 Remark. 5 A(3, 4) 6. (A half of the 12 vertices of a regular icosahedron gives such an example. We want to determine A(n, t) for general n and t, but most cases are open.

17 Let (So, c t > 0.) c t = Min{Q t (x) 1 x 1}. 16 Apply Delsarte s method (LP-method) to F (s) = c t + Q t (s) on 1 s 1. (Note that F (s) 0, for 1 s 1. ) Then we get Theorem (B-O-T) If Y is a design of harmonic index t on S n 1, then Y b t (= b t,n ) := 1 + Q t(1) c t.

18 Proof. Calculate (x,y) Y Y F (x y) in two ways. 17 (i) F (x y) = (c t + Q t (x y)) = c t Y 2 (x,y) Y Y (x,y) Y Y (Since (x,y) Y Y Q t(x y) = 0 if Y is a design of harmonic index t.) (ii) F (x y) Y F (1). (x,y) Y Y (Since F (s) 0 on 1 s 0.) So, c t Y 2 Y F (1). Hence Y 1 + Q t(1) c t (= b t ).

19 18 In the above proof, note that, by using Addition formula, (x,y) Y Y Q i (x y) = = m i j=1 ( y Y (x,y) Y Y m i ( f j (x)f j (y)) j=1 f j (x)) 2 0. and = 0, if and only if y Y f(y) = 0, for f(x) Harm i(r n ). (So, (x,y) Y Y Q t(x y) = 0 if and only if Y is a design of harmonic index i.) Here, m j = dim(harm i (R n )), and {f 1, f 2,..., f mi } is any orthonormal basis of Harm i (R n ).

20 Values of b t = b t,n for designs of harmonic index t on S n 1 19 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 t = t = t = t = t = t = t = t = t =

21 Spherical designs of harmonic index LetY be a design of harmonic index 4 on S n 1. Then (n + 1)(n + 2) Y. 6 If Y = (n+1)(n+2) 3 6, then I(Y ) = {±α} with α = n+4 where Q 4 (α) gives the minimum value of Q 4 (x). So, Y gives a set of equiangular lines ( (n+1)(n+2) 6 lines) with inner product α = 3 n+4. Okuda-Yu (2015) showed that there exist no such set of equiangular lines (by using SDP-method). So, there is no tight spherical designs of harmonic index 4 if n 3. (Cf. Wei-Hsuan Yu s talk.)

22 What happens if t = 2e is fixed, and n? 21 t = 4 = b 4 = (n+1)(n+2) n2. 1 t = 6 = b 6 20(2+ 10) n3. t = 8 = b 8 1 ( 3,000) n4... t = 2e = b 2e (const)n e. (Note that this constant is explicitly calculable.) If Y is a tight design of harmonic index 2e, then I(Y ) = {±α}, and so, Y n(n+1) 2 (absolute bound). So, there are no tight designs of harmonic index 2e, if e 3, in general.

23 What happens for b 2e,n, if n is fixed and e? 22 lim e b 2e,n are, for n = 3, 4,..., 10: (6), (10), (15), (21) (38), (36), (45) (55), (resectively), where inside the parentheses denote the value n(n+1) 2. So, lim e b 2e,n is far bigger than n(n+1) 2, except for some small n, say n 6. It would be a very interesting open question, what are the values of A(3, 2e) (or A(4, 2e)) for large e. This implies that tight designs of harmonic index 2e do not exist in general (i.e. for large t), unless n is very small.

24 Let T = {t 1, t 2..., t l } be a set of natural numbers. Y S n 1, 0 < Y < is called a spherical design of harmonic index T, if y Y f(y) = 0 for f(x) Harm i(r n ) with i T. 23 (Spherical t-design is a design of harmonic index {1, 2,..., t}.) t = 8 {1, 2,..., 8}-design {2, 8}-design {4, 8}-design {6, 8}-design {8}-design Y 1 24 n4 Y n4 for {2, 8} Y n4 for {4, 8} Y 1 ( 3,000) n4

25 (V) spherical null t-design 24 Let Y S n 1, 0 < Y < w : Y R 0 Y is called a spherical null t-design y Y w(y)f(y) = 0 for f(x) = f(x 1,..., x n ) of degree t y Y w(y)f(y) = 0 for f(x) Harm i(r n ) with i = 0, 1,..., t

26 25 What are Fisher type lower bound for null spherical t-designs on S n 1? (working with Francis Campena) Let A(n, t) = min{ Y Y is a null spherical t-design} What are A(n, t)? Conjecture A(n, t) = 2(t + 1), n 2, t Partial results A(n, t) (t + 1), n 2, t A(2, t) = (t + 1), t A(3, 2) = 6, A(n, 2) = 6, n 2 (First open case A(3, 3), 7 A(3, 3) 8)

27 Replace S n 1 by association schemes (II) weighted combinatorial t-designs (cf. talks of Da Zhao and Zongchen Chen) (I) combinatorial t-designs designs in J(v, k) (III) relative t-designs on J(v, k) (Delsarte (1977), Pairs of vectors...) (cf. talk of Yan Zhu) (II) weighted t-designs in H(n, 2) (I) t-designs in H(n, 2) (III) relative t-designs on H(n, 2) (Delsarte (1977)) (=weighted regular t-wise balanced designs) (cf. talks by Yan Zhu and Lihang Hou) 26

28 Fisher type lower bouds (I) (II) = Y m e, for t = 2e where m i = m i + m i 1 + { ( + m 1 + m 0 v ) ( m i = rank(e i ) = i v i 1) for J(v, k) ( n i) for H(n, 2) 27 (III) = Y m e + m e 1 + m e p+1 for t = 2e study of tight relative t-designs on H(n, q) and J(v, k) (cf. talks by Yan Zhu and Lihang Hou)

29 Thank You