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) R esearch C enter for P ure and A pplied M athematics G raduate S chool of I nformation S ciences 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 Q n 1:1 [ n ] := { 1 , 2 , . . . , n } ( n ∈ N ) → 2 [ n ] { 0 , 1 } n ← Q n := 2 [ n ] = { x : x ⊆ [ n ] } : the n -cube 1010 · · · 0 ← → { 1 , 3 } 1110 · · · 0 ← → { 1 , 2 , 3 } 011 = { x ∈ 2 [ n ] : | x | = k } � [ n ] � 001 111 k 101 ∅ ̸ = Y ⊂ 2 [ n ] 010 000 ω : Y → R > 0 110 100 Definition (Delsarte (1977)) “weighted” regular t -wise balanced design ( Y, ω ) : a relative t -design def ⇐ ⇒ ∃ λ 1 , λ 2 , . . . , λ t ∈ R > 0 s.t. for i = 1 , 2 , . . . , t , � � [ n ] � ω ( y ) = λ i ( ∀ z ∈ ) i z ⊂ y ∈ Y Hajime Tanaka Tight relative t -designs in hypercubes August 18, 2019 2 / 24
Delsarte’s design theory � [ n ] � � � Φ := k : Y ∩ ̸ = ∅ k | Φ | = 1 ( t -designs) | Φ | ⩾ 2 (relative t -designs) Delsarte (1973) Delsarte (1977) spherical t -designs Euclidean t -designs Delsarte–Goethals–Seidel Neumaier–Seidel (1988) (1977) Hajime Tanaka Tight relative t -designs in hypercubes August 18, 2019 3 / 24
Tight relative t -designs � [ n ] � k : Y ∩ � ̸ = ∅ � Recall Φ = . k Theorem (Bannai–Bannai (2012); Xiang (2012)) ( Y, ω ) : a relative 2 e -design e ⩽ k ⩽ n − e ( ∀ k ∈ Φ) Then � n � n n � � � � | Y | ⩾ + + · · · + . e − 1 e −| Φ | +1 e Fisher-type inequality Definition � n def � n n � � � � ( Y, ω ) : tight ⇐ ⇒ | Y | = + + · · · + e − 1 e −| Φ | +1 e 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 � [ n ] � ( Y, ω ) : a tight 2 e -design ⊂ k take complement: k ↔ n − k Then Y induces an e -class Q -polynomial association scheme. 1 The polynomial 2 � − ξ, − e, e − n � � ψ k � e ( ξ ) := 3 F 2 � 1 ∈ R [ ξ ] � k − n + 1 , 1 − k of 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 � − ξ, − e, e + α + β + 1 � � � Q e ( ξ ; α, β, N ) = 3 F 2 � 1 ∈ R [ ξ ] . � α + 1 , − N They are orthogonal polynomials if α, β > − 1 or α, β < − N . Remark ψ k e ( ξ ) = Q e ( 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 aaaaaaaaaaaaaaaaa take complement: ℓ ↔ n − ℓ , m ↔ n − m Theorem Φ = { ℓ, m } where e ⩽ ℓ < m ⩽ n − ℓ � [ n ] � [ n ] � � ( Y, ω ) : a tight relative 2 e -design ⊂ ⊔ ℓ m Then � e +1 e � Y induces a coherent configuration of type . 1 e +1 The polynomial 2 � − ξ, − e, e − n − 1 � � ψ ℓ,m � ( ξ ) := 3 F 2 � 1 ∈ R [ ξ ] e � m − n, − ℓ of degree e has only integral zeros. Remark ψ ℓ,m ( ξ ) = Q e ( ξ ; m − n − 1 , − m − 1 , ℓ ) e Hajime Tanaka Tight relative t -designs in hypercubes August 18, 2019 8 / 24
A tool in the proof | Φ | = 1 ( t -designs) | Φ | = 2 (relative t -designs) Bose–Mesner algebra Terwilliger algebra (commutative) (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 ψ ℓ,m ( ξ ) e 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 ψ ℓ,m ( ξ ) e Example The existence of tight relative 4 -designs with the following feasible parameters were left open: n ℓ m ξ √ 1 37 9 16 14 (71 ± 337) √ 1 37 9 21 14 (55 ± 337) √ 1 37 16 28 14 (55 ± 337) √ 1 14 (71 ± 37 21 28 337) √ 1 41 15 16 26 (237 ± 1569) √ 1 41 15 25 26 (153 ± 1569) √ 1 26 (153 ± 41 16 26 1569) √ 1 41 25 26 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 2 e -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 � � − e/ 2 , − ( e − 1) / 2 � − 1 � H e ( η ) = (2 η ) e · 2 F 0 � ∈ R [ η ] . � − η 2 Q e ( ξ ; α, β, N ) ≈ H e ( η ) for appropriate limit | α | , | β | , N → ∞ Hajime Tanaka Tight relative t -designs in hypercubes August 18, 2019 13 / 24
Askey scheme Hypergeometric orthogonal polynomials 1 Wilson Racah Continuous Continuous Hahn Dual Hahn dual Hahn Hahn Meixner Pseudo - Jacobi Meixner Krawtchouk Jacobi Pollaczek Laguerre Bessel Charlier Hermite 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 ψ k e ( ξ ) = Q e ( ξ ; k − n, − k − 1 , k − 1) , Bannai showed n = 2 k + 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 = 2 k + 1 . e ( ξ ) = a 0 ξ e + a 1 ξ e − 1 + · · · + a e − 1 ξ + a e . Write ψ k Then a 1 /a 0 , a 2 /a 0 , . . . , a e /a 0 ∈ 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 = (2 b + 1)(2 b + 3) · · · (2 b + 2 a − 1) has a prime factor > 2 a + 1 , except for the following cases: a = 1 and s = 3 i ( i ⩾ 2) . a = 2 and s = 25 · 27 ; 2 1 2 a + 1 = 5 2 a + 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 ⇒ 183 · 185 · 187 · 189 = 3 4 · 5 · 7 · 11 · 17 · 37 · 61 a = 4 = ⇒ 201 · 203 · 205 · 207 · 209 = 3 3 · 5 · 7 · 11 · 19 · 23 · 29 · 41 · 67 a = 5 = 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) ψ k e ( ξ ) = Q e ( ξ ; k − n, − k − 1 , k − 1) ≈ H e ( η ) for n, k → ∞ n = 2 k + 1 for n, k ≫ 0 Case Φ = { ℓ, m } ( e ⩽ ℓ < m ⩽ n − ℓ ) ψ ℓ,m ( ξ ) = Q e ( ξ ; m − n − 1 , − m − 1 , ℓ ) ≈ H e ( η ) for n, ℓ, m → ∞ e n = 2 m 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 = 2 k + 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 = 2 m 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) ∃ e 0 = e 0 ( δ ) > 0 such that: Fix e ⩾ e 0 and c > 0 . 1 Then only finitely many such tight relative 2 e -designs with 2 ℓ < c · n 1 − δ . up to scalar multiple of ω Remark We can take e 0 ( δ ) = exp(3 /δ ) . Hajime Tanaka Tight relative t -designs in hypercubes August 18, 2019 20 / 24
ℓ A non-existence result for the case | Φ | = 2 Fix e ⩾ e 0 ( δ ) and set c = 1 , 000 = 10 3 . (This is not a precise graph.) 10 3 ⋅ n 1 − δ n only finitely many here Hajime Tanaka Tight relative t -designs in hypercubes August 18, 2019 21 / 24
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