Association schemes with multiple Q -polynomial structures Jianmin - - PowerPoint PPT Presentation

Association schemes with multiple Q-polynomial structures

Jianmin Ma Hebei Normal University Shanghai Jiaotong University 9/27/2014

Association Scheme

Let R0, R1, . . . , Rd be binary relations on set X: (a) R0 = {(x, x)|x ∈ X} (b) R0 ∪ R1 ∪ · · · ∪ Rd = X × X, Ri ∩ Rj = ∅ (c) RT

i

= Rj for some j (0 ≤ j ≤ d) (d) a △ counting for pk

ij

z

j

• x

i

• k

y

◮ Commutative AS ⇔ pi jk = pi kj. ◮ Symmetric AS ⇔ Ri = RT i

for all i.

Examples: Polygon, Hamming

◮ Polygons: ◮ Hamming: H(d,q) on Qd with alphabets Q of order q.

(x, y) ∈ Ri if x and y differ in i positions

◮ Why Hamming? Error-correcting Codes

◮ Hamming codes ◮ Golay codes ◮ Reed-Muller codes, . . .

Johnson J(d, n)

Johnson: J(d, q) on d-subsets of Q. (S, T) ∈ Ri if |S ∩ T| = d − i Why Johnson? Designs on subsets

Examples: finite groups

◮ (Transitive group) Association schemes are a combinatorial

generalization of transitive permutation groups: if the group G is transitive on Ω, the orbits of G on Ω2 form an association scheme. J(3,7): Group S7 acts transitively on 3-subsets Ω of [1..7]: R1 = {(A, B) | |A ∩ B| = 2}. Graph (Ω, R1) has diameter 3: A i ∼ B ⇔ |A ∩ B| = 3 − i.

◮ (Group case) Let G be a finite group with conjugacy classes

C0 = {1}, C1, . . . , Cd (x, y) ∈ Ri if yx−1 ∈ Ci.

Definition of association scheme

Symmetric association scheme: X = (X, {R0, R1, . . . , Rd}). Adjacency matrices: A0, . . . , Ad. A0 = I, AT

i = Ai,

A0 + · · · + Ad = J, AiAj =

d

• k=0

pk

ijAk.

Bose-Mesner algebra: A = C[A0, . . . , Ad]. Note: Ai ◦ Aj = δijAi → A has a second basis: E0, . . . , Ed, primitive idempotents. E0 = |X|−1J, E T

i

= Ei, E0 + · · · + Ed = I, Ei ◦ Ej = |X|−1

k

• k=0

qk

ijEk.

Eigenvalues/eigenmatrices: (A0, A1, . . . , Ad) = (E0, E1, . . . , Ed)P (E0, E1, . . . , Ed) = |X|−1(A0, A1, . . . , Ad)Q

Definition of association scheme

Symmetric association scheme: X = (X, {R0, R1, . . . , Rd}). Adjacency matrices: A0, . . . , Ad. A0 = I, AT

i = Ai,

A0 + · · · + Ad = J, AiAj =

d

• k=0

pk

ijAk.

Bose-Mesner algebra: A = C[A0, . . . , Ad]. Note: Ai ◦ Aj = δijAi → A has a second basis: E0, . . . , Ed, primitive idempotents. E0 = |X|−1J, E T

i

= Ei, E0 + · · · + Ed = I, Ei ◦ Ej = |X|−1

k

• k=0

qk

ijEk.

Eigenvalues/eigenmatrices: (A0, A1, . . . , Ad) = (E0, E1, . . . , Ed)P (E0, E1, . . . , Ed) = |X|−1(A0, A1, . . . , Ad)Q

For a 3-cube: V = {000} ∪ {001, 010, 100} ∪ {011, 110, 101} ∩ {111} d(x, y) = Hammming distance Ai = distance i matrix. Row 1 of A1 =

• 1

1 1

• A2 = 3I + 0 · A + 2A2 + 0 · A3 ⇒A2 ∈ I, A, A2,

A2 ∈ I, A, A2, A2 / ∈ I, A, AA2 = 0 · I + 2A + 0 · A2 + 3A3 ⇒A3 ∈ I, A, A2, A3 A3 ∈ I, A, A2, A3, A3 / ∈ I, A, A2 AA3 = A2 ⇒A4 ∈ I, A, A2, A3 ⊂ I, A, A2, A3

• 1. Each Aj is a polynomial in A of deg j
• 2. (V , R1) is a distance-regular graph

Polynomial structure, distance-regular graph

Example J(3,7): Ω= 3-subsets Ω of [1..7]: R1 = {(A, B) | |A ∩ B| = 2}. Graph Γ = (Ω, R1) has diameter 3: A h ∼ B ⇔ |A ∩ B| = 3 − h. Take O7 := (Ω, R3), the Odd graph of diameter 3.

Definition

If there are polynomials vi of degree i such that Ai = vi(A1), then A0, A1, . . . Ad is called P-polynomial structure: i.e., A1Aℓ = cℓ−1Aℓ−1 + aℓAℓ + bℓ+1Aℓ+1. In this case, (X, A1) is a distance-regular graph. J(3,7) has two P-polynomial structures: (Ω, R1), (Ω, R3).

Polynomial structure, distance-regular graph

Example J(3,7): Ω= 3-subsets Ω of [1..7]: R1 = {(A, B) | |A ∩ B| = 2}. Graph Γ = (Ω, R1) has diameter 3: A h ∼ B ⇔ |A ∩ B| = 3 − h. Take O7 := (Ω, R3), the Odd graph of diameter 3.

Definition

If there are polynomials vi of degree i such that Ai = vi(A1), then A0, A1, . . . Ad is called P-polynomial structure: i.e., A1Aℓ = cℓ−1Aℓ−1 + aℓAℓ + bℓ+1Aℓ+1. In this case, (X, A1) is a distance-regular graph. J(3,7) has two P-polynomial structures: (Ω, R1), (Ω, R3).

Q-polynomial structure

Definition

Suppose scheme X has primitive idempotents: E0, . . . , Ed. If there are polynomials v∗

i of degree i such that Ei = v∗ i (E1) with

entry-wise product, then E0, E1, . . . Ed is called Q-polynomial structure: i.e., E1 ◦ Ej = |X|−1(c∗

j−1Ej−1 + a∗ j Ej + b∗ j+1Ej+1).

No combinatorial interpretation for Q-polynomial structure! There are association schemes with more than one such structures.

Q-polynomial structure

Definition

Suppose scheme X has primitive idempotents: E0, . . . , Ed. If there are polynomials v∗

i of degree i such that Ei = v∗ i (E1) with

entry-wise product, then E0, E1, . . . Ed is called Q-polynomial structure: i.e., E1 ◦ Ej = |X|−1(c∗

j−1Ej−1 + a∗ j Ej + b∗ j+1Ej+1).

No combinatorial interpretation for Q-polynomial structure! There are association schemes with more than one such structures.

Main object

Problem: Study association schemes with multiple Q- or P-polynomial structures. How many P-polynomial structure can a scheme have?

◮ If a scheme is from an ordinary n-gon, it has φ(n)

P-polynomial structures, where φ(n) = |{k | (k, n) = 1, 1 ≤ k ≤ n}|.

◮ What about otherwise?

Multiple P-polynomial structures

• Ei. Bannai
• Et. Bannai

Theorem (Bannai-Bannai, 1980)

If A0, A1, . . . , Ad is a P-polynomial structure for an association scheme X which is not an ordinary polygon, then any second such structure must be one of the following: (I) A0, A2, A4, A6, . . . , A5, A3, A1; (II) A0, Ad, A1, Ad−1, A1, A2, Ad−2, A3, Ad−3, . . . ; (III) A0, Ad, A2, Ad−2, A4, Ad−4, . . . , Ad−3, A3, Ad−1, A1 ; (IV) A0, Ad−1, A2, Ad−3, A4, Ad−5, . . . , A3, Ad−2, A1, Ad ; (X, R) admits at most two P-polynomial structures.

Multiple P- or Q- structures for DRG

In the book of Bannai and Ito (1984), they proved for (P and Q)-polynomial schemes X with Q-structure E0, E1, . . . , Ed, and intersection numbers ai, bi, ci. If X is not a polygons, diameter d ≥ 34 and has another Q-structure E ′

0, E ′ 1, . . . , E ′ d, then ◮ the possible form of E ′ 0, E ′ 1, . . . , E ′ d were determined; ◮ the possible form of ai, bi, ci in terms of Leonard’s theorem. ◮ X can have at most two P-polynomial structures. ◮ All eigenvalues are rational integers.

They remarked similar assertions hold if X has instead another P-polynomial structure.

P-Polynomial structures

In their 1980 paper, Ei. Bannai and Et. Bannai asked

• Q1. whether similar result holds for AS with multiple

Q-polynomial structures?

• Q2. determine all association schemes with two P-structures;
• Q3. whether type III (A0, Ad, A2, . . . ) can hold for larger d ≥ 4?

◮ Hiroshi Suzuki answered Q3 in 1993: d ≤ 4

(published in 1996).

◮ He answer Q1 in 1996, published in 1998. See

below.

• H. Suzuki

P-Polynomial structures

In their 1980 paper, Ei. Bannai and Et. Bannai asked

• Q1. whether similar result holds for AS with multiple

Q-polynomial structures?

• Q2. determine all association schemes with two P-structures;
• Q3. whether type III (A0, Ad, A2, . . . ) can hold for larger d ≥ 4?

◮ Hiroshi Suzuki answered Q3 in 1993: d ≤ 4

(published in 1996).

◮ He answer Q1 in 1996, published in 1998. See

below.

• H. Suzuki

Dickie and Terwilliger’s attack on 2Q DRGs

Theorem (Dickie 1995)

Let Γ be a distance regular graph with diameter d ≥ 5 and valency k ≥ 3. Then Γ has two Q-polynomial structures if and only if Γ is

• ne of the following:

(i) the cube H(d, 2) with d even; (ii) the half cube 1

2H(2d + 1, 2);

(iii) the folded cube ˜ H(2d + 1, 2); (iv) the dual polar graph on [2A2d−1(q)], where q ≥ 2 is a prime power. The Academy Award for “Scientific and Technical Achievement”

For the primary design (Perry Kivolowitz) and for the development (Garth Dickie) of the algorithms, for the shape-driven warping and morphing subsystem of the Elastic Reality Special Effects System.

Dickie and Terwilliger’s attack on 2Q DRGs

Theorem (Dickie 1995)

Let Γ be a distance regular graph with diameter d ≥ 5 and valency k ≥ 3. Then Γ has two Q-polynomial structures if and only if Γ is

• ne of the following:

(i) the cube H(d, 2) with d even; (ii) the half cube 1

2H(2d + 1, 2);

(iii) the folded cube ˜ H(2d + 1, 2); (iv) the dual polar graph on [2A2d−1(q)], where q ≥ 2 is a prime power. The Academy Award for “Scientific and Technical Achievement”

For the primary design (Perry Kivolowitz) and for the development (Garth Dickie) of the algorithms, for the shape-driven warping and morphing subsystem of the Elastic Reality Special Effects System.

Twice Q-polynomial DRGs with diameter 3 or 4

• 1. Any distance-regular graph of diameter 2 is a strongly regular

graph, which has two P- and two Q- polynomial structures.

• 2. Determine distance-regular graphs with two Q-polynomial

structures and diameter 3 or 4.

• 3. Know examples:

◮ Diameter d = 4:

H(d, 2), 1

2H(2d + 1, 2), ˜

H(2d + 1, 2), [2A2d−1(q)] and Hadamard graphs of order 2γ with intersection array {2γ, 2γ − 1, γ, 1; 1, γ, 2γ − 1, 2γ} with γ = 1 or γ a positive even integer.

◮ Diameter d = 3:

1 2H(2d + 1, 2), ˜

H(2d + 1, 2), [2A2d−1(q)] and a distance-regular graph with intersection array (A) {k, µ, 1; 1, µ, k}

• r

(B) {k, k − 1, k − µ; 1, µ, k} where µ < k − 1.

Note (A) is bipartite, and (B) is antipodal, called a Taylor graph.

2Q DRG, diameter 4

Theorem (Koolen-M 2014,main theorem 1)

Let Γ denote a twice Q-polynomial distance- regular graph with diameter d = 4 and valency at least 3. Then Γ is one of the following: (i) the cube H(d,2) ; (ii) the half cube 1/2H(2d + 1, 2);

• J. Koolen

(iii) the folded cube ˜ H(2d + 1, 2); (iv) the dual polar graph on [2A2d−1(q)], where q ≥ 2 is a prime power; (v) a Hadamard graph of order 2γ with intersection array {2γ, 2γ − 1, γ, 1; 1, γ, 2γ − 1, 2γ} with γ = 1 or γ a positive even integer.

Theorem (Ma 2014, main theorem 2)

Let Γ be a distance regular graph with diameter d = 3 and valency k ≥ 3. Then Γ has two Q-polynomial structures if and only if Γ is at least one of: (i) the half cube 1/2H(2d + 1, 2); (ii) the folded cube ˜ H(2d + 1, 2); (iii) the dual polar graph on [2A5(q)] with q ≥ 2 a prime power; (iv) a distance-regular graph with intersection array (A) {k, µ, 1; 1, µ, k}

• r

(B) {k, k − 1, k − µ; 1, µ, k} where µ < k − 1.

Proof of main theorems

Abbr: Q2-DRG for any distance-regular graph with exact two Q structures.

Theorem

Let Ω be a distance-regular graph with intersection numbers aj. Suppose that E0, E1, . . . , Ed is a Q-polynomial structure with Krein parameters a∗

j . If Γ is a Q2-DRG, then the following implications

hold: (i) a∗

1 = 0

= ⇒ a∗

2 = · · · = a∗ d−1 = 0;

(ii) a1 = 0 = ⇒ a2 = · · · = ad−1 = 0; (iii) a∗

1 = 0

= ⇒ a∗

r = 0 for all r (2 ≤ r ≤ d − 1);

(iv) a1 = 0 = ⇒ ar = 0 for all r (2 ≤ r ≤ d − 1). Note that (iv) holds for any DRG.

Case argument for d = 3 or 4

[a∗

1 = a∗ d = 0]. Dual bipartite distance-regular graphs with

diameter at least 3 were determined Dickie and Terwilliger. [a∗

1 = 0 = a∗ d]. Dickie proved the following result:

Lemma (Dickie)

Let Ω be distance-regular graph with diameter d ≥ 4. Suppose that E0, E1, . . . , Ed and ˜ E0, ˜ E1, . . . , ˜ Ed are Q-polynomial structures for Ω with Krein parameters a∗

i and ˜

a∗

i , respectively. If a∗ 1 = 0 and

˜ a∗

1 = 0, then E1 = ˜

Ed, Ed = ˜ E1 and d = 4.

• – One of the main result of Koolen-M is to show that a∗

1˜

a∗

1 = 0

for d = 4. – In fact, for d = 3, a∗

1 = 0 = a∗ d and ˜

a∗

1 = 0 = ˜

a∗

d can not both

hold (to be proved later).

Remaining cases

[a∗

1 = 0 = a∗ d]

— The case d = 4 was determined by Dickie. — The case d = 3. Terwilliger (1988) determined the parameters for Γ when it is neither 1

2H(2d + 1, 2) nor ˜

H(2d + 1, 2). I will show this case is [2A5(r)]. [a∗

1 = 0 = a∗ d]. Then ˜

a∗

1 = 0 by the above result and the above

cases apply.

The rest is devoted to the proofs of two results for d = 3.

Theorem

Let Γ denote a distance-regular graph with diameter 3. Then Γ can not have two Q-polynomial structures that both have a∗

1 = 0 = a∗ 3.

Theorem

Let Γ be a Q-polynomial DRG with diameter 3 and a∗

1 = a∗ 2 = 0 = a∗

• 3. Then Γ is 1

2H(7, 2), ˜

H(7, 2), or [2A5(r)] for some prime power r. Now we switch to the blackboard presentation.