Tight relative t -designs on two shells in hypercubes, and Hahn and - - PowerPoint PPT Presentation

Tight relative t-designs on two shells in hypercubes, and Hahn and Hermite polynomials

Hajime Tanaka

(joint work with Eiichi Bannai, Etsuko Bannai, and Yan Zhu) Research Center for Pure and Applied Mathematics Graduate School of Information Sciences Tohoku University

August 18, 2019 G2D2

Hajime Tanaka Tight relative t-designs in hypercubes August 18, 2019 1 / 24

Relative t-designs in the n-cube Qn

[n] := {1, 2, . . . , n} (n ∈ N) Qn := 2[n] ={x : x ⊆ [n]} : the n-cube {0, 1}n

1:1

← → 2[n]

1010 · · · 0 ← → {1, 3} 1110 · · · 0 ← → {1, 2, 3}

000 100 010 001 101 011 110 111

[n]

k

• = {x ∈ 2[n] : |x| = k}

∅ ̸= Y ⊂ 2[n] ω : Y → R>0

Definition (Delsarte (1977))

“weighted” regular t-wise balanced design

(Y, ω) : a relative t-design

def

⇐ ⇒ ∃λ1, λ2, . . . , λt ∈ R>0 s.t. for i = 1, 2, . . . , t,

• z⊂y∈Y

ω(y) = λi (∀z ∈ [n]

i

• )

Hajime Tanaka Tight relative t-designs in hypercubes August 18, 2019 2 / 24

Delsarte’s design theory

Φ :=

• k : Y ∩

[n]

k

• ̸= ∅
• |Φ| = 1 (t-designs)

Delsarte (1973)

spherical t-designs

Delsarte–Goethals–Seidel (1977)

|Φ| ⩾ 2 (relative t-designs)

Delsarte (1977)

Euclidean t-designs

Neumaier–Seidel (1988)

Hajime Tanaka Tight relative t-designs in hypercubes August 18, 2019 3 / 24

Tight relative t-designs

Recall Φ =

• k : Y ∩

[n]

k

• ̸= ∅
• .

Theorem (Bannai–Bannai (2012); Xiang (2012))

(Y, ω) : a relative 2e-design e ⩽ k ⩽ n − e (∀k ∈ Φ) Then |Y | ⩾ n

e

• +

n

e−1

• + · · · +
• n

e−|Φ|+1

• .

Fisher-type inequality

Definition

(Y, ω) : tight

def

⇐ ⇒ |Y | = n

e

• +

n

e−1

• + · · · +
• n

e−|Φ|+1

• Hajime Tanaka

Tight relative t-designs in hypercubes August 18, 2019 4 / 24

The tight case with |Φ| = 1

Theorem (Delsarte (1973), Wilson–Ray-Chaudhuri (1975))

Φ = {k} where e ⩽ k ⩽ n/2

take complement: k ↔ n − k

(Y, ω) : a tight 2e-design ⊂ [n]

k

• Then

1

Y induces an e-class Q-polynomial association scheme.

2

The polynomial ψk

e(ξ) := 3F2

−ξ, −e, e − n k − n + 1, 1 − k

• 1
• ∈ R[ξ]
• f degree e has only integral zeros.

a Hahn polynomial

Hajime Tanaka Tight relative t-designs in hypercubes August 18, 2019 5 / 24

The tight case with |Φ| = 1

Remark

Ito (1975), Bremner (1979): only 2 examples for e = 2 Peterson (1977): none for e = 3 Bannai (1977): only finitely many, for each e ⩾ 5 Dukes–Short-Gershman (2013): none for e = 5, 6, 7, 8, 9 Xiang (2018): none for e = 4

Hajime Tanaka Tight relative t-designs in hypercubes August 18, 2019 6 / 24

The Hahn polynomials

α, β ∈ R, N ∈ N The Hahn polynomial of degree e (e = 0, 1, . . . , N) is Qe(ξ; α, β, N) = 3F2 −ξ, −e, e + α + β + 1 α + 1, −N

• 1
• ∈ R[ξ].

They are orthogonal polynomials if α, β > −1 or α, β < −N.

Remark

ψk

e(ξ) = Qe(ξ; k − n, −k − 1, k − 1)

Hajime Tanaka Tight relative t-designs in hypercubes August 18, 2019 7 / 24

The tight case with |Φ| = 2

Theorem

aaaaaaaaaaaaaaaaa take complement: ℓ ↔ n − ℓ, m ↔ n − m

Φ = {ℓ, m} where e ⩽ ℓ < m ⩽ n − ℓ (Y, ω) : a tight relative 2e-design ⊂ [n]

ℓ

• ⊔

[n]

m

• Then

1

Y induces a coherent configuration of type e+1

e e+1

• .

2

The polynomial ψℓ,m

e

(ξ) := 3F2 −ξ, −e, e − n − 1 m − n, −ℓ

• 1
• ∈ R[ξ]
• f degree e has only integral zeros.

Remark

ψℓ,m

e

(ξ) = Qe(ξ; m − n − 1, −m − 1, ℓ)

Hajime Tanaka Tight relative t-designs in hypercubes August 18, 2019 8 / 24

A tool in the proof

|Φ| = 1 (t-designs)

Bose–Mesner algebra (commutative)

|Φ| = 2 (relative t-designs)

Terwilliger algebra (non-commutative)

For a preceding study, see also

• E. Bannai, E. Bannai, S. Suda, and H. Tanaka,

On relative t-designs in polynomial association schemes,

• Electron. J. Combin. 22 (2015) #P4.47.

Hajime Tanaka Tight relative t-designs in hypercubes August 18, 2019 9 / 24

Use of ψℓ,m

e

(ξ)

Example

Bannai–Bannai–Zhu (2017) found four tight relative 4-designs for n = 22: n ℓ m ξ 22 6 7 3, 5 22 6 15 1, 3 22 7 16 1, 3 22 15 16 3, 5 The zeros ξ are integers!!

Hajime Tanaka Tight relative t-designs in hypercubes August 18, 2019 10 / 24

Use of ψℓ,m

e

(ξ)

Example

The existence of tight relative 4-designs with the following feasible parameters were left open: n ℓ m ξ 37 9 16

1 14(71 ±

√ 337) 37 9 21

1 14(55 ±

√ 337) 37 16 28

1 14(55 ±

√ 337) 37 21 28

1 14(71 ±

√ 337) 41 15 16

1 26(237 ±

√ 1569) 41 15 25

1 26(153 ±

√ 1569) 41 16 26

1 26(153 ±

√ 1569) 41 25 26

1 26(237 ±

√ 1569) The zeros ξ are irrational, thus proving the non-existence!!

Hajime Tanaka Tight relative t-designs in hypercubes August 18, 2019 11 / 24

Bannai’s result revisited (|Φ| = 1)

Theorem (Bannai (1977))

Fix e ⩾ 5. Then only finitely many non-trivial tight 2e-designs. Extend the result to the case |Φ| = 2 !!

Hajime Tanaka Tight relative t-designs in hypercubes August 18, 2019 12 / 24

Ingredients in Bannai’s proof, part 1

The Hermite polynomial of degree e (e = 0, 1, 2, . . .) is He(η) = (2η)e · 2F0 −e/2, −(e − 1)/2 −

• − 1

η2

• ∈ R[η].

Qe(ξ; α, β, N) ≈ He(η) for appropriate limit |α|, |β|, N → ∞

Hajime Tanaka Tight relative t-designs in hypercubes August 18, 2019 13 / 24

Askey scheme

Hypergeometric orthogonal polynomials1

Racah Jacobi Meixner Krawtchouk Meixner

• Pollaczek

Continuous Hahn Hahn Dual Hahn Continuous dual Hahn Wilson Laguerre Charlier Hermite Pseudo Jacobi Bessel

1 taken from: R. Koekoek, P

. A. Lesky, and R. F. Swarttouw, Hypergeometric orthogonal polynomials and their q-analogues, Springer-Verlag, Berlin, 2010. Hajime Tanaka Tight relative t-designs in hypercubes August 18, 2019 14 / 24

Ingredients in Bannai’s proof, part 1

Recall Φ = {k} (e ⩽ k ⩽ n/2). Applying the limit process to ψk

e(ξ) = Qe(ξ; k − n, −k − 1, k − 1),

Bannai showed n = 2k + 1 with only finitely many exceptions.

Hajime Tanaka Tight relative t-designs in hypercubes August 18, 2019 15 / 24

Ingredients in Bannai’s proof, part 2

Assume n = 2k + 1. Write ψk

e(ξ) = a0ξe + a1ξe−1 + · · · + ae−1ξ + ae.

Then a1/a0, a2/a0, . . . , ae/a0 ∈ Z. Bannai showed this is impossible by using the following:

Theorem (Schur (1929))

a, b ∈ N (a < b) Then the product of a consecutive odd integers s = (2b + 1)(2b + 3) · · · (2b + 2a − 1) has a prime factor > 2a + 1, except for the following cases:

1

a = 2 and s = 25 · 27;

2

a = 1 and s = 3i (i ⩾ 2).

2a + 1 = 5 2a + 1 = 3

Hajime Tanaka Tight relative t-designs in hypercubes August 18, 2019 16 / 24

Ingredients in Bannai’s proof, part 2

Example

a = 3 = ⇒ 91 · 93 · 95 = 3 · 5 · 7 · 13 · 19 · 31 a = 4 = ⇒ 183 · 185 · 187 · 189 = 34 · 5 · 7 · 11 · 17 · 37 · 61 a = 5 = ⇒ 201 · 203 · 205 · 207 · 209 = 33 · 5 · 7 · 11 · 19 · 23 · 29 · 41 · 67

Hajime Tanaka Tight relative t-designs in hypercubes August 18, 2019 17 / 24

Extension to the case |Φ| = 2 ?

Concerning part 1:

Case Φ = {k} (e ⩽ k ⩽ n/2)

ψk

e(ξ) = Qe(ξ; k − n, −k − 1, k − 1) ≈ He(η) for n, k → ∞

n = 2k + 1 for n, k ≫ 0

Case Φ = {ℓ, m} (e ⩽ ℓ < m ⩽ n − ℓ)

ψℓ,m

e

(ξ) = Qe(ξ; m − n − 1, −m − 1, ℓ) ≈ He(η) for n, ℓ, m → ∞ n = 2m for n, ℓ, m ≫ 0 No control over ℓ !!

Hajime Tanaka Tight relative t-designs in hypercubes August 18, 2019 18 / 24

Extension to the case |Φ| = 2 ?

Concerning part 2:

Case Φ = {k} (e ⩽ k ⩽ n/2)

n = 2k + 1 for n, k ≫ 0 by part 1. Set a = ⌊e/2⌋, b = k − e + 1 in Schur’s theorem. Integrality fails, and thus proves non-existence.

Case Φ = {ℓ, m} (e ⩽ ℓ < m ⩽ n − ℓ)

n = 2m for n, ℓ, m ≫ 0 by part 1. Set a = ⌊e/2⌋, b = m − e + 1 in Schur’s theorem. Integrality may hold if ℓ behaves badly!!

Hajime Tanaka Tight relative t-designs in hypercubes August 18, 2019 19 / 24

A non-existence result for the case |Φ| = 2

Recall Φ = {ℓ, m} (e ⩽ ℓ < m ⩽ n − ℓ).

Theorem

∀δ ∈ (0, 1) ∃e0 = e0(δ) > 0 such that:

1

Fix e ⩾ e0 and c > 0.

2

Then only finitely many such tight relative 2e-designs with ℓ < c · n1−δ.

up to scalar multiple of ω

Remark

We can take e0(δ) = exp(3/δ).

Hajime Tanaka Tight relative t-designs in hypercubes August 18, 2019 20 / 24

A non-existence result for the case |Φ| = 2

Fix e ⩾ e0(δ) and set c = 1, 000 = 103.

n

ℓ

103 ⋅ n1−δ

(This is not a precise graph.)

• nly finitely many here

Hajime Tanaka Tight relative t-designs in hypercubes August 18, 2019 21 / 24

A non-existence result for the case |Φ| = 2

Fix e ⩾ e0(δ) and set c = 10, 000 = 104.

n

ℓ

104 ⋅ n1−δ

(This is not a precise graph.)

• nly finitely many here

Hajime Tanaka Tight relative t-designs in hypercubes August 18, 2019 22 / 24

A non-existence result for the case |Φ| = 2

Fix e ⩾ e0(δ) and set c = 100, 000 = 105.

n

ℓ

105 ⋅ n1−δ

(This is not a precise graph.)

• nly finitely many here

Hajime Tanaka Tight relative t-designs in hypercubes August 18, 2019 23 / 24

A variation of Schur’s theorem

Proposition

∀δ ∈ (0, 1) ∃a0 = a0(δ) > 0 such that:

1

Fix N ∋ a ⩾ a0 and c > 0.

2

Then for all but finitely many pairs (b, d) ∈ N2 with d < c · b1−δ, the product of a consecutive odd integers (2b + 1)(2b + 3) · · · (2b + 2a − 1) has a prime factor > 2a + 1 which is not a prime factor of (2d + 1)(2d + 3) · · · (2d + 2a − 1).

Hajime Tanaka Tight relative t-designs in hypercubes August 18, 2019 24 / 24