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Developments of Algebraic Combinatorics

a personal view

Eiichi Bannai

Shanghai Jiao Tong University

Sept 13, 2013

Talk at Egawa 60 conference

Tokyo University of ScienceTokyo Japan

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Finite group theory in 1960’s (−→ 1970’s)

• The attempt to classify finite simple groups had been started.( Feit-Thompson, M. Suzuki, Gorenstein, Aschbacher, K. Harada, . . . . . . . . . )

• Many new finite simple groups were discovered.(Janko, Higman-Sims, Conway, Fisher, . . . . . . . . . )

Finite simple groups

Zp cyclic groups of order pAn alternating groupssimple groups of Lie typesporadic simple groups (26 of them)

• Finite permutation groups were very much sterdied.

• Multiply transitive permutation groups. (Wielandt, M. Hall, . . . )• Rank 3 permutation groups. (Higman-Sims, . . . )• Primitive permutation groups (D. G. Higman, . . . )

(permutation groups of maximum diameter = distance transitive graphs)

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I was fascinated with finite group theory.

The development of finite group theory was very dynamic !

Japanese group theorists at that time

in USA in Japan

M. Suzuki (Univ. Illinois) Tokyo (Iwahori,. . . )

N. Ito (Chicago Circle) Osaka (Nagao,. . . )

K. Harada (Ohio State) Hokkaido (Tsuzuku,. . . )

4Ohio State University

Harada 1968 1972 -

Bannai -1974 1976 1978 -1989

Egawa -1977 1981

Y. Egawa: PhD in 1980 (adviser K. Harada)Standard component problem (in pure group theory)

In late 1970’s, there was a move from“ group theory ” to “algebraic combinatorics ”

5rank 3 perm. group −→ strongly regular graph

G y Ω transitively (v, k, λ, µ)

α ∈ Ω

v = |Ω| = |V |,k = |Γ1(α)|

Gα has 3 orbits

α, Γ1(α), Γ2(α)

SRG of type (10, 3, 0, 1)v, k, λ, µ

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distance transitive graph

(permutation gp. of max. diameter)

x, y, z, w ∈ V

d(x, y) = d(z, w)

⇒ ∃g ∈ G

s.t. z = xg, w = yg

distance regular graph

0 1

k a1 c2

b1 a2 cd−1

b2 ad−1 cd

bd−1 ad

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transitive perm. group

of subdegrees

1, k, k(k − 1), k(k − 1)2,

· · · , k(k − 1)d

−→ Moore graph

0 1k 0 1

k − 1 0k − 1

10 1

k − 1 k − 1

Hoffman-Singleton

d = 2 =⇒ k = 2, 3, 7 or 57

k = 57 =⇒ still open

d ≥ 3, k ≥ 3 =⇒ Non-existence.

8More generally,

transitive perm. proup −→ association scheme

multiplicity free trans.

perm. group −→ com. association scheme

a perm. group −→ coherent configuration

(not necessarily transitive)

9Association schemes

X = (X, Ri0≤i≤d).

Ri ⊂ X × X

1. R0 = (x, x) | x ∈ X,

2. R0, R1, . . . , Rd gives a partition of X × X,

3. for each i ∈ 0, 1, . . . , d, tRi = Ri′

with some i′ ∈ 0, 1, . . . , d,

where tRi = (y, x) | (x, y) ∈ Ri,

4. For i, j, k ∈ 0, 1, . . . , d,

|z ∈ X | (x, z) ∈ Ri, (z, y) ∈ Rj| = pki,j.

A0, A1, . . . , Ad are the adjacency matrices

for the relations R0, R1, . . . , Rd

10We want to classify distance transitive graphs

and distance regular graphs

Question 1.

Which parameters are possible for distance regular graphs ?

Question 2.

Can we characterize distance regular graphs with

given set of parameters ?

Y. Egawa: Characterization of H(n, q) by the parameters,

J.C.T.(A), 1981.(Characterization of H(n, 2) by the parameters was done earlier

by H. Enomoto)

Y. Egawa: Association schemes of quadratic forms,

J.C.T.(A), 1985.

11The set X of all quadratic forms of degree n over Fq

We can get a structure of distance regular graphs by combining

several relations together.

This means that there is an advantage of “considering associa-

tion schemes ” over “considering just the action of the group”.

After this work, Y. Egawa (also H. Enomoto) started to study

graph theory.

12What is Algebraic Combinatorics?E. Bannai-T. Ito: Algebraic Combinatorics, I

(Benjamin/Cummings, 1984).

Our philosophy: ”group theory without groups”.

Let X = (X, Ri(0≤i≤d)) be a symmetric association scheme.

Let ⟨A0, A1, . . . , Ad⟩ = ⟨E0, E1, . . . , Ed⟩ be the Bose-Mesner

algebra.

X is a P-poly. ⇐⇒ ∃vi(x) = poly. of degree exactly i,

assoc. scheme 0 ≤ i ≤ d , such that Ai = vi(A1)

⇐⇒ (X, R1) is a distance regular graph

X is a Q-poly. ⇐⇒ ∃v∗i (x) = poly. of degree exactly i,

assoc. scheme 0 ≤ i ≤ d , such that |X|Ei = v∗i (|X|E1)

13Our main theme:Classify P- and Q-polynomial association schemes.• X = P- and Q-poly. assoc. schme

=⇒ both vi(x) (0 ≤ i ≤ d) and v∗i (x) (0 ≤ i ≤ d)

are orthgonal polynomials (they are mutually related).

=⇒ They are expressed by Ashkey-Wilson polynomials,

or by their special cases or limiting cases. (Leonard 1982)q = ±1 q −→ 1 q −→ −1

Case I Case II Case III

Recently, this class of poly-nomials have very much stud-ied in the theory og orthogonalpolynomials. (Tsujimoto-Vinet-Zhedanov [20] 2012)

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T. Ito-P. Terwilliger determined irreducible representations of

Terwilliger algebra (for case I).

Now we should come back to the study of P- and Q- polynomial

association schemes using their results. (For example, charac-

terizations by parameters.)

15Next steps

multiplicity free perm. groups(= Gelfand pairs )

−→ commutative assoc. schemes(Cf. Martin-Tanaka [19] 2009)

any finite groups −→ group association schemes(commutative assoc. schemes)

pairs of a finite group and a max-imal subgroup

−→ primitive association schemes

finite simple groups −→ primitive group assoc. schemes(commutative)

We would like to study finite simple groups from the view pointof association schemes and algebraic combinatorics.At present, this is not a realistic problem. But we should try,and should not give up.

Another source of Algebraic Combinatorics 16(Delsarte Theory)

What are codes and designs ?

M = a set (finite or infinite metric space)

e.g.(V

k

)(= set of a k elements subset of V ),

Sn−1 = (x1, . . . , xn) | x21 + · · · + x2

n = 1

Johnson association scheme J(v, k) is defined on(V

k

).

Hamming association scheme H(n, q) is defined as follows.X = F n, where F is a finite set of q elements.For x = (x1, . . . , xn), y = (y1, . . . , yn) ∈ X we define(x, y) ∈ Ri ⇐⇒ |ν | xν = yν| = i, for i = 0, 1, . . . , n

17Purpose of coding theory

Find subset Y ⊂ M

such that

mind(x, y); x, y ∈ Y, x = yis maximum

M = H(n, q) =⇒usual coding theory.

M = Sn−1 =⇒optimal spherical code

Purpose of design theory

Find subset Y ⊂ M which approximates the whole

space M well.

combinatorial t-design

(t-(v, k, λ)- design, 1 ≤ t ≤ k ≤ v)

X =(V

k

)Y ⊂ X is a t-(v, k, λ)- design

⇐⇒ |y ∈ Y | T ⊂ y| = λ (constant) for ∀T ⊂(V

t

).

18M = Sn−1

spherical t-design

Y ⊂ Sn−1

1|Sn−1|

∫Sn−1 f(x)dσ(x) = 1

|Y |∑

y∈Y f(y)

∀polynomial f(x) = f(x1, . . . , xn) of degree ≤ t.

regular polyhedron # of vertices t-design

tetrahedron 4 2

cube 8 3

octahedron 6 3

icosahedron 12 5

dodecahedron 20 5240 roots of type E8(⊂ R8) is a 7-design

196560 min. vectors of Leech lattice (⊂ R24) is an 11-design

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Delsarte (1973) unified the study of codes and designsin the frame work of assoc. schemes.

An algebraic approach to the theory of association schemes of

the coding theory [10] (1973)

P-poly. assoc. scheme −→ error correcting e-codes.

(H(n, q) −→ usual coding theory)

Q-poly. assoc. scheme −→ t-designs.

(J(v, k) =(V

k

)−→ usual design theory)

Delsare-Goethals-Seidel [12] (1977) studied codes and designs

on the sphere Sn−1, in a similar way as codes and designs were

studied in association schemes.

A natural way to find t -(v, k, λ) designs is to take a finite 20subgroup G ⊂ Sv, and take the orbit Y = yG of G on y ∈

(Vk

).

Also, a natural way to find spherical t-designs is to take a finite

subgroup G ⊂ O(n) and take an orbit Y = yG of G on y ∈ Sn−1.

However, it seems that if such Y is a t-design (i.e., t-(v, k, λ)

design or spherical t-design on Sn−1 with n ≥ 3), then t must

be bounded.

So, we cannot construct t-(v, k, λ) designs or spherical t-designs

for large t, from groups.

However, t-(v, k, λ) design exist for any t (Teirlinck, 1987) (forsome v, k and λ), and spherical t-designs in Sn−1 exist for any tand n. (Seymour-Zaslavsky, 1984). See also, recent arXiv papers:(1) Kuperberg-Lovett-Peled, arXiv:1302.4295,(2) Fazeli-Lovett-Vardy, arXiv:1306.2088,(3) Bondarenko-Radchenko-Viazovska, Ann of Math.(2013), and

arXiv:1303.5991.

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Note that Delsarte’s definition of t-design for any Q-polynomial

association scheme X = (X, Ri(0≤i≤d)) is given as follows.

Let Y be a subset of X, and let φY be the characteristic vector

(function) of Y . Then, Y is a t-design, if and only if

EiφY = 0, for i = 1, 2, . . . , t.

Then we have the following Fisher type inequalities. (For sim-

plicity, we assume t = 2e.)

22• If t = 2e and Y be a 2e-design in

(Vk

)(i.e., in Johnson assoc.

scheme J(v, k), or more generally for any Q-polynomial assoc.

scheme X = (X, Ri0≤i≤d)), then

|Y | ≥ me + me−1 + · · · + m1 + m0,

where mi = rank of Ei.

Namely, the RHS is(v

e

)for J(v, k),

and is equal to(v

e

)+

( ve−1

)+ · · · +

(v0

)for H(n, 2),

since mi =(v

i

)−

( vi−1

)for J(v, k),

and mi =(v

i

), for H(n, 2).

• If t = 2e and Y is a spherical 2e-design in Sn−1, then

|Y | ≥ me + me−1 + · · · + m1 + m0 =(n−1+e

e

)+

(n−1+e−1e−1

),

where mi =(n−1+i

i

)−

(n−1+i−2i−2

).

( Delsarte-Goethals-Seidel, 1977).

23So, we are interested in t-designs of possible smallest cardinality

|Y |. (Those which satisfy the equality in Fisher type inequality

are called tight t-designs.The classification of tight 2e-designs in J(v, k).• e = 1 =⇒ symmetric 2-designs. (many examples).• e = 2 =⇒ 4-(23, 7, 1) design or 4-(23, 16, 52) design.(Enomoto-Ito-Noda, 1979.)• e = 3 =⇒ Non-existence (Peterson, 1977)• e ≥ 4 =⇒ for each e there are only finitely many tight 2e-designs, (B, 1977)• 5 ≤ e ≤ 9 =⇒ Non-existence (Dukes and Short-Gershmen, 2013),• e = 4 =⇒ Non-existence (Z. Xiang, 2012 (unpublished))(Open for e ≥ 10.)

The classification of tight spherical t-designs in Sn−1 (we assume n ≥ 3).• If there is a tight t -design =⇒ t ∈ 1, 2, 3, 4, 5, 7, 11 .

(Bannai-Damerell, 1979/80, Bannai-Sloane, 1981)Now, the classification is open only for t = 4, 5, 7.(Cf. Bannai-Munemasa-Venkov (2004, Algebra i Analiz),Nebe-Venkov (2012, Algebra i Analiz).)

24Generalizations of the concept of t-designs

There are many generalizations of t-(v, k, λ) designs and spher-

ical t-designs.

1. t-designs in other (Q-polynomial) association schemes,

and t-designs in other compact symmetric spaces of rank 1

(i.e. projective spaces over R, C, H, O.) That is,

Y ⊂ X, EiφY = 0, for i = 1, 2, . . . , t,

or

Y ⊂ M,1

|M |

∫M

f(x)dσ(x) =1

|Y |∑y∈Y

f(y).

for appropriate f(x).

252. Allow weight function w : Y −→ R>0

Namely, let (Y, w) satisfy:

either ∑T⊂y∈Y

w(y) = λi (constant), for all T ∈ V (i)

for i = 1, 2, . . . , t. (in t-(v, k, λ) design case),

or1

|Sn−1|

∫Sn−1

f(x)dσ(x) =∑y∈Y

w(y)f(y)

for any polynomials f(x) of degree at most t

(in spherical t-design case).

263. Allow different sizes of blocks (in t-(v, k, λ) design case), or

different radii of spheres (in spherical case). Namely, either

Y ⊂ V (k1) ∪ V (k2) ∪ · · · ∪ V (kp)

or

Y ⊂ Sn−1(r1) ∪ Sn−1(r2) ∪ · · · ∪ Sn−1(rp).

27Remark. The generalization of two steps 2 and 3of t-(v.k.λ) design (=t-design in J(v, k),) is called

”(weighted) regular t-wise balanced design”,

and this concept is equivalent to the concept of

”(weighted) relative t-design” in H(n, 2)

in the sense of Delsarte (1977).

On the other hand, the generalization of two steps 2 and 3of spherical t-designs are called Euclidean t-designs (on p shells).

28Fisher type lower bound are known for relativet-designs in H(n, 2) and Euclidean t-designs.

(Here, we assume t = 2e for simplicity.)

• If t = 2e and (Y, w) is a (weighted) relative 2e-design in H(n, 2)

and if Y is on p-shells (i.e. if Y ⊂ V (k1) ∪ V (k2) ∪ · · · ∪ V (kp)),

then

|Y | ≥ me + me−1 + · · · + me−p+1,

where mi =(v

i

). (Z. Xiang (JCT(A),2012).)

• If t = 2e and (Y, w) is an Euclidean 2e-design in Rn on p-shells,

then

|Y | ≥ me + me−1 + · · · + me−p+1

where mi =(n−1+i

i

)−

(n−1+i−2i−2

). (Muller, 1970’s).

29• The concept of Euclidean t-design was defined by Neumaier-Seidel (1988)and also studied by Delsarte-Seidel (1989). (Similar concepts have been stud-ied in other areas, e.g. numerical analysis, statistics, etc.)We have studied ”tight” Euclidean t-designs extensively for the last 10 years.(There are many interesting examples, and some classification results, thoughthey are still partial.)

• The concept of relative t-design (in Q-polynomial) association schemes) wasintroduced by Delsarte (1977) [13]. The study of tight relative t-designs hasjust started. See for our recent results,Bannai-Bannai-Suda-Tanaka [6] (arXiv: 1303.7163),Bannai-Bannai-Bannai [4] (arXiv:1304.5769).

I believe that now it is the time to start the study of tight relative t-designs

in general (Q-polynomial) association schemes more systematically.

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Thank You