Design theory from the viewpoint of algebraic combinatorics

Transcription

1 Design theory from the viewpoint of algebraic combinatorics Eiichi Bannai Shanghai Jiao Tong University May 27, 2017, at 9SHCC This talk is based on the paper: Eiichi Bannai, Etsuko Bannai, Hajime Tanaka, Yan Zhu, Design theory from the view point of algebraic combinatorics, Graphs and Comb. 33 (2017)

2 Design theory is to find a small subset Y which approximates the given whole space M. M = ( V k) and M = S n 1. Combinatorial t-design: Let M = ( V k), where V = v. Then Y M is called a t-(v, k, λ) design, if #{y Y z y} = λ (constant) for all z ( V ) t. History of combinatorial t-designs. Kirkman, Steiner, Witt, Fisher, Bose, Hughes, Hall, Ray-Chaudhuri, Wilson, Delsarte, Brouwer, Teirlinck, Keevash, etc.... 2

3 2 Delsarte (1973): An algebraic approach to the theory of association schemes of coding theory, gave (i) The concept of Q-polynomial association scheme, and defined the concept of algebraic t-design for any Q-polynomial association scheme. Namely, for the Bose-Mester algebra A = A 0, A 1,..., A d = E 0, E 1,..., E d, E i ϕ Y = 0 for all i = 1, 2,..., t, where ϕ Y is the characteristic vector of Y. (ii) The algebraic t-design in Johnson association scheme J(v, k) is equal to t-(v, k, λ) design. 3

4 Let M = S n 1 be the unit sphere in R n. Definition (Delsarte-Goethals-Seidel, 1977). For a positive integer t, a subset Y S n 1, 0 < Y <, is called a spherical t-design on S n 1, if 1 S n 1 S n 1 f(x)dσ(x) = 1 Y x Y f(x) holds for any polynomials f(x) = f(x 1, x 2,..., x n ) of degree t. This condition is equivalent to the condition f(x) = 0, for any f(x) Harm i (R n ), x Y for all i = 1,..., t. 4

5 There are many similarities between the study of combinatorial t-designs and that of spherical t-designs. There are some generalizations of the t-design concept, for both combinatorial and spherical designs. (i) Consider the weight function w : Y R >0, and consider the weighted t-design (Y, w). (ii) Allow the blocks of different sizes for combinatorial t-designs (= relative t-designs in binary Hamming association schemes H(d, 2)), and also allow different radii of spheres for spherical designs (=Euclidean t-designs). 5

6 (iii) Consider T -designs for any subset T {1, 2,..., d} for combinatorial designs (or more generally for a Q-polynomial association scheme), and for any subset T {1, 2,...} for spherical designs. (We call such T -design a design of harmonic index T.) Namely, E i ϕ Y = 0, for any i T (for case H(d, 2) or for a Q-poly association scheme case), and f(x) = 0 x Y for any f Harm i (R n ) with i T (for the spherical case). 6

7 Note again that there are strong similarities between the study of combinatorial t-designs and that of spherical t-designs: the space of functions on M = ( V k) and M = S n 1. the space spanned by column vectors of E i Harm i (R n ). spherical functions and addition formulas. Hahn polynomials Gegenbauer polynomials. Fisher type inequalities (for t = 2e). Y ( v) ( Y n 1+e ) ( + n 1+e 1 ). e e e 1 (Petrenjuk, Ray-Chaudhuri and Wilson) (Delsarte-Goethals- Seidel) 7

8 The classification of tight t-designs (t 4). (Enomoto-Ito-Noda, Peterson, Bannai, Best, Hong, Dukes and Short-Gershman, Z.Xiang, etc.) (Bannai-Damerell, Bannai-Sloane, Makhnev, Bannai-Munemasa- Venkov, Nebe-Venkov, etc.) t 20 are open (but finitely many for each t 4) Only t = 4, 5, 6 are open. We want to discuss three generalizations of the t-design concept (i), (ii), (iii) mentioned already. (1) (i) + (ii)= Relative t-design (in H(d, 2)) (Delsarte, 1977: Pairs of vectors in the space of association scheme.) 8

9 Allow several different sizes of blocks Let Y be a subset of ( ) ( V r 1 V ) ( r 2 V ) r p, where V = d. Y is called a regular t-wise balanced design if #{y Y z y} = λ i (constant) for all z ( V ) i for each fixed i = 1, 2,..., t. Note that Y is regarded as a subset of the binary Hamming association scheme H(d, 2). Such Y is called a (constant weight) relative t-design on X = H(d, 2). A relative t-design (Y, w) on X is also defined. Namely, w(y) = λ i (constant) for all z ( V ) i z y Y for each fixed i = 1, 2,..., t More generally, relative t-design is defined for any Q-polynomial association scheme X = (X, {R i } 0 i d ) as follows. 9

10 Let x 0 X be fixed. Y X (or more generally (Y, w)) is called a relative t-design on X with respect to x 0, if E i ϕ Y (or E i ϕ (Y,w) ) and E i ϕ {x0 } are linearly dependent for all i = 0, 1,..., t. (2) (i) + (ii) = Euclidean t-design (a generalization of spherical t-design.) (Neumaier-Seidel, 1988, Delsarte-Seidel, 1989.) Euclidean t-design is a two step generalization of spherical t- design. Step 1: Allow weight function w : Y R >0. Namely, 1 f(x)dσ(x) = w(x)f(x) S n 1 S n 1 x Y for any polynomials f(x) = f(x 1, x 2,..., x n ) of degree t. 10

11 Step 2: Allow several shells of spheres (p shells). Namely, let Y S n 1 (r 1 ) S n 1 (r 2 ) S n 1 (r p ), and let w : Y R >0. Then (Y, w) in a Euclidean t-design (on p shells), if and only if p w(y S n 1 (r i )) i=1 S n 1 (r i ) S n 1 (r i ) f(x)dσ i (x) = x Y w(x)f(x) for any polynomials f(x) = f(x 1, x 2,..., x n ) of degree t. 11

12 Fisher type lower bound are obtained for Euclidean t-designs, and also for relative t-designs on H(d, 2). Here we assume t = 2e for simplicity. Formula for odd t exists but is more complicated. If (Y, w) is a Euclidean 2e -design on p shells, then we have Y m e + m e m e p+1, where m i = ( n 1+i) ( i n 1+i 2 ) i 2 = dim Harmi (R n ). If Y is a relative 2e-design of H(d, 2) on p shells, then Y m e + m e m e p+1, where m i = ( d i) = rank of Ei, for X = H(d, 2). 12

13 We tried to work on the classification problems of tight Euclidean t-designs, in particular on small number of shells (e.g. for p = 2.) Jointly with Etsuko and other coauthors, we published about papers on the classification problem on tight Euclidean t-designs. The first paper was: On Euclidean tight 4-designs, J. Math. Soc. Japan (2006) and the most recent one is: Tight t-designs on two concentric spheres, Moscow J. Comb. and Number Theory (2014). See also: A survey on spherical designs and algebraic combinatorics on the sphere, Europ. J. Comb. (2009). (So far, the obtained results are still partial, and many problems are yet to be answered.) 13

14 At one point (around 2009), I noticed that a similar definition as Euclidean t-design and the theory of Euclidean t-design can be generalized to association schemes. I was very much excited. However, Hajime Tanaka immediately pointed out to me that this is exactly the concept of relative t-design by Delsarte (1977) that we defined already in a few slides before. (Until that time, I was completely unaware of the existence of the concept and the Delsarte s paper (1977): On pairs of vectors in association schemes.) 14

15 In 2010 (Sept-Dec), Etsuko and I visited Hebei Normal University, and read the paper of Delsarte: Pairs of vectors... in a weekly seminar with Suogang Gao and his group. Jointly with Zengti Li, we studied Fisher type lower bound for relative t-designs in binary Hamming association scheme H(d, 2). But we could not succeed completely, Cf. Li-Bannai-Bannai: Graphs and Comb. (2015). In February 2011, I joined SJTU, and had a weekly informal seminar, and proposed this problem. The answer was finally obtained by Z. Xiang, Fisher type inequality for regular t-wise balanced designs, JCT(A),

16 Namely, if Y is a relative 2e -design of H(d, 2) on p shells, then Y m e + m e m e p+1, where m i = ( d i) rank of Ei for X = H(d, 2), as already mentioned. 16

17 Although the concept of relative t-designs on H(d, 2) (and on other Q-polynomial association schemes) is older than that of Euclidean t-designs, it seems tight t-designs were not much studied until we started the study of it, after through the study of tight Euclidean t-designs. Etsuko and I wrote several papers on tight relative t- designs joint with other co-authors, including Zengti Li, Hideo Bannai, Yan Zhu, Sho Suda and Hajime Tanaka, in the last 5 years. And we hope to study more on this subject. (There are lots of interesting open problems here yet to be studied.) 17

18 Generally we expect that if Y is a tight relative 2edesign on a Q-polynomial association scheme X = (X, {R i } 0 i d ), and Y X r1 X r2 X rp, then Y m e + m e m e p+1 where m i = rank(e i ) of X (Note that Y m e, if p = 1). 18

19 Delsarte (1977) used relative t-design to interpret Assmus- Mattason Theorem. (Assmus-Mattson (1969), Delsarte (1977), Tanaka (2009)) Let X = (X, {R i } 0 i d ) be a Q-polynomial association scheme. For x 0 X, let X r = {x X (x 0, x) R r }. Let Y be a t-design on X on a union of p shells, i.e. Y X r1 X r2 X rp. Then each Y ri = Y X ri, (i = 0, 1,..., p) is a relative (t p + 1)-design on X with respect to x 0. Remark 1: Our recent work (Bannai-Bannai-Tanaka-Zhu, 2017) shows that the assumption Y is a t-design is replaced by a weaker condition that Y is a relative t-design (for x 0 ). Then we still can get the same conclusion. 19

20 Remark 2: Originally, relative t-design was defined with respect to the Q-structure. We can also define relative t-designs with respect to the P-structure. (Cf. Bannai-Bannai-Suda-Tanaka: On relative t-designs in polynomial association schemes, Electronic J. Comb, 2015.) For X = H(d, q), these two concepts are the same. But for other P-and Q- polynomial association schemes, these two concepts are actually different. We propose to study relative t-designs Y of any P-and Q-polynomial association scheme X with Y X r, one shell of X. We propose to call such Y as the t-design on the one shell X r. 20

21 Here, I will not discuss the details of our recent work on relative t-designs of certain association schemes. (Some was already talked by Yan Zhu in the first day of this conference 9SHCC.) For our recent work, please see Zhu-Bannai-Bannai: Tight relative 2-designs on two shells in Johnson association scheme. Discrte Math (2015). Bannai-Bannai-Zhu: Relative t-designs in binary Hamming association scheme H(n, 2), Designs Codes and Crypt. (2017). Bannai-Bannai-Tanaka-Zhu: Design theory from the view point of algebraic combinatorics. Graphs and Comb. (2017). Bannai-Zhu: Tight t designs on one shall of Johnson association schemes, submitted. 21

22 Conclusions (Speculations). We should study tight relative t-designs Y on Q-polynomial association schemes more systematically. I believe that the time is ripe for this study and we get similar results for tight t design in Q-polynomial association schemes. (Mostly non-existence results for large t. But small t cases are also interesting as there are possibilities of the existence of some examples related to some classical combinatorial concept.) In particular we should study tight t-design Y on X with Y X r, for each of Q-structure and P-structure, and for each P-and Q-polynomial association scheme X. 22

23 This kind of study might be useful, (i) for the study of general commutative association schemes, and (ii) for the classification problem of P-and Q-polynomial association schemes. 23

24 Consider the generalization (iii) of t-design concept = T -design (= harmonic index T -design). Let T {1, 2, 3,...}. Then Y S n 1 is called a spherical T -design, if x Y f(x) = 0 for any f Harm i (R n ) with i T. Note that if T = {1, 2,..., t}, then T -design is an ordinary t-design. Previously, only T -designs with T = {1, 2,..., t}, or T = {2, 4,..., 2e}, have been studied. However, we can actually study for other T. Bannai-Okuda-Tagami, J. Approx. Theory (2015), Okuda-Yu, Europ. J. Comb. (2016), Zhu-Bannai-Bannai-Kim-Yu, Electronic J. Comb. (2017). 24

25 Bannai-Okuda-Tagami (2015) started to study Fisher type lower bound for spherical {2e}-designs. So, we can consider tight spherical {2e}-designs. Theorem Let Y be a harmonic index 4-design on S n 1. if Y is a 4-design, then Y n(n + 3)/2. if Y is a {4}-design (i.e., harmonic index 4-design), then Y (n + 1)(n + 2)/6. Moreover, if Y = (n+1)(n+2) 6, then {x y x, y Y, x y} = {±α} with α = 3 n+4. So, Y gives a set of equiangular lines ( (n+1)(n+2) lines) 6 with inner product α = 3. n+4 25

26 Okuda-Yu (Europ. J. Comb., 2016) showed that there exist no such set of equiangular lines (by using SDPmethod). So, there is no tight spherical designs of harmonic index 4 if n 3. Main Results of Zhu-Bannai-Bannai-Kim-Yu (2017) (Assume n 3, t = 2e 4.) (i) If Y is a design of harm. index t = 2e, then Y cn e (if n ). (c = c(e) depends only on e, and is calculable.) Note that if Y is a spherical 2e-design, then Y 1 e! ne. 26

27 (ii) Tight spherical designs of harmonic index 6 and 8 do not exist. (iii) We get the asymptotic non-existence of tight spherical designs of harmonic index 2e, for general e 3. (iv) We get the non-existence of tight spherical designs of harmonic index T with T = {8, 4}, {8, 2}, {8, 6}, {12, 8, 4}, {10, 6, 2}, etc. Exactly speaking, for the case of {12, 8, 4}, the following three cases are not yet settled: n = 9, 13, 16 and Y = 285, 1311, 3315 (respectively). 27

28 Results (Bannai-Okuda-Tagami) If Y is a {4}-design, then Y 5. Moreover, those with Y = 5 are determined. That is, they are regular pentagons with the fixed size (corresponding two positive zeros of Legendre polynomials of degree 4, or they are those any of their points are replaced by the antipodal points. If Y is a {6}-design, then 4 min of Y 7. If Y is a {8}-design, then 4 min of Y 6. (Any half of 12 vertices of an icosahedron (taking one for each antipodal points) is an {8}-design, as well as {14}-design on S 2. If Y is a {2e}-design, then 4 min of Y??. 28

29 Recently, I proposed the following problems in the class. Problem (i). Consider 240 root vectors of type E 8. We choose one for each pair of antipodal two vectors. Then, can we choose so that sum of these 120 vectors is the zero vector? Problem (ii). Consider minimum vectors of Leech lattice. We choose one for each pair of antipodal two vectors. Then, can we choose so that sum of these vectors is the zero vector? 29

30 The asswer to Problem (i) and Problem (ii) are both positive. Moreover, for we can answer when this property holds and when this property does not hold, for all root systems of type A n, D n, and E n. Note that any half of E 8 root vectors (normalized) is a {6, 4, 2}-design. Any half of minimum vectors of Leech lattice (normalized) is a {10, 8, 6, 4, 2}-design. Problem (i) means whether it can become a {6, 4, 2, 1}- design, and Problem (ii) means whether it can become a {10, 8, 6, 4, 2, 1}-design. We can discuss many related problems for T -designs. Paper on this topic is in preparation by Bannai, Da Zhao, Lin Zhu, Yan Zhu, Yinfeng Zhu. 30

31 Note that Y S n 1 is called a tight frame if and only if Y is a {2}-design. Theorem. (Barg-Glazyrin-Okoudjou-Yu, LAA, 2015) If Y S n 1 is a tight frame (i.e., {2}-design) and 2- distance set (with inner products α, β) then we get either (i) α 2 β 2, and then Y is a spherical embedding of a SRG, or a shifted spherical embedding of a SRG, or (ii) α 2 β 2, and Y is a tight equiangular frame. (In this case an SRG is attached, but Y is not directly related the spherical embedding of that SRG.) 31

32 We are now studying the following situation. Let Y S n 1 be a {4, 2, 1}-design and 2-distance set. (Then Y is necessarily a spherical embedding of a SRG.) Then, we can determine all possible parameters of the SRG, although when they actually exist or not remains as an open problem. The paper is in preparation by Ei Bannai, Et Bannai, Ziqing Xiang, Wei-Hsuan Yu, Yan Zhu. 32

33 Anyway, there are many reasons why we want to study harmonic index T -designs. For further details, please see: Zhu-Bannai-Bannai-Kim-Yu, On spherical designs of harmonic indices, (Electronic J. Comb. 2017), Zhu-Bannai-Bannai-Ikuta-Kim, Harmonic index designs in binary Hamming schemes, (Graphs and Comb. to appear). 33

34 Thank you very much 34