Upload
others
View
4
Download
0
Embed Size (px)
Citation preview
1
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
2
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)
3
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
6
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
7
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)
14
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
19
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.
21
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.