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Association schemes and orthogonal polynomials
Hajime Tanaka
Research Center for Pure and Applied MathematicsGraduate School of Information Sciences
Tohoku University
Novemver 25, 2017International Workshop on Bannai–Ito Theory
Hajime Tanaka Orthogonal polynomials Novemver 25, 2017 1 / 33
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Notation from “Algebraic Combinatorics I” (1984)
X = (X, {Ri}Di=0) : a (symmetric) association schemeA0, A1, . . . , AD : the adjacency matricesE0, E1, . . . , ED : the primitive idempotentsP = (pj(i))
Di,j=0, Q = (qj(i))
Di,j=0 : the eigenmatrices
Aj =
D∑i=0
pj(i)Ei, Ej =1
|X|
D∑i=0
qj(i)Ai (0 ⩽ j ⩽ D)
Set kj = pj(0), mj = qj(0) (0 ⩽ j ⩽ D).
valency multiplicity
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pj(i)
kj=
qi(j)
mi(0 ⩽ i, j ⩽ D) Askey–Wilson duality
D∑i=0
mi pj(i) pℓ(i) = δj,ℓ|X|kj orthogonality relation
D∑i=0
ki qj(i) qℓ(i) = δj,ℓ|X|mj orthogonality relation∗
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Suppose X is P - & Q-polynomial.
Set θi = p1(i), θ∗i = q1(i) (0 ⩽ i ⩽ D).∃fj(x), f∗
j (x) ∈ R[x] s.t. deg fj(x) = deg f∗j (x) = j and
fj(θi) =pj(i)
kj, f∗
j (θ∗i ) =
qj(i)
mj(0 ⩽ i, j ⩽ D).
1 f0(x) = f∗0 (x) = 1
2 xfj(x) = cjfj−1(x) + ajfj(x) + bjfj+1(x) 3-term recurrence
3 xf∗j (x) = c∗jf
∗j−1(x) + a∗jf
∗j (x) + b∗jf
∗j+1(x) 3-term recurrence∗
4 fj(θi) = f∗i (θ
∗j ) Askey–Wilson duality
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Constructions (univariate, terminating)
scheme polynomials referenceHamming Krawtchouk Delsarte (1973)Johnson Hahn, dual Hahn Delsarte (1973)
Grassmannq-Hahn,
Delsarte (1976)dual q-Hahn
bilinear forms affine q-Krawtchouk Delsarte (1978)
dual polarq-Krawtchouk,
Stanton (1980)dual q-Krawtchouk
polygons q-Racah Bannai–Ito (1984)Odd Bannai-Ito Bannai–Ito (1984)
Merit: orthogonality follows automatically.Demerit: does not cover full parameter range.[Example] Bannai-Ito polynomials have 4 parameters r1, r2, s, s∗ besides D.
Hajime Tanaka Orthogonal polynomials Novemver 25, 2017 5 / 33
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Leonard’s theorem (1982)
Let {fj(x)}Dj=0, {f∗j (x)}Dj=0 ⊂ R[x], where D ∈ N ∪ {∞}, satisfy
1 f0(x) = f∗0 (x) = 1
2 xfj(x) = cjfj−1(x) + ajfj(x) + bjfj+1(x)
3 xf∗j (x) = c∗jf
∗j−1(x) + a∗jf
∗j (x) + b∗jf
∗j+1(x)
4 fj(θi) = f∗i (θ
∗j ) for some {θi}Di=0, {θ∗i }Di=0 ⊂ R
Then they are the Askey–Wilson polynomials
4ϕ3
(q−j , s∗qj+1, q−y, sqy+1
r1q, r2q, r3q
∣∣∣∣ q; q) (0 ⩽ j ⩽ D)
in the variable x = θ0 + h(1− qy)(1− sqy+1)q−y, where
ss∗ = r1r2r3 (r3 = q−D−1 if D <∞),
q-Racah polynomialsor some of their limits in the Askey scheme.
Hajime Tanaka Orthogonal polynomials Novemver 25, 2017 6 / 33
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Askey scheme
q-Hypergeometric orthogonal polynomials, part 1‡
Askey-Wilson
Continuousq -Hahn Big q -Jacobi
q -Meixner-
PollaczekBig
q -LaguerreLittle
q -JacobiAl-Salam
-Chihara
Continuousdual q -Hahn
Continuousq -Jacobi
Continuousbig q -Hermite
Continuousq -Laguerre
Littleq -Laguerre q -Laguerre
Stieltjes-
WigertContinuousq -Hermite
‡ taken from: R. Koekoek, P. A. Lesky, and R. F. Swarttouw, Hypergeometric orthogonal polynomials and their
q-analogues, Springer-Verlag, Berlin, 2010.
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Askey scheme
q-Hypergeometric orthogonal polynomials, part 2‡
q -Racah
Quantumq -Krawtchouk q -Krawtchouk Affine
q -KrawtchoukDual
q -Krawtchouk
Dualq -Hahnq -Hahn
q -Meixner
Big q -Jacobi
q -CharlierAl-Salam
- Carlitz I
Al-Salam-
Carlitz IIq -Bessel
Discreteq -Hermite I
Discreteq -Hermite II
‡ taken from: R. Koekoek, P. A. Lesky, and R. F. Swarttouw, Hypergeometric orthogonal polynomials and their
q-analogues, Springer-Verlag, Berlin, 2010.
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Askey scheme
Hypergeometric orthogonal polynomials‡
Racah
Jacobi Meixner KrawtchoukMeixner
-Pollaczek
ContinuousHahn Hahn Dual HahnContinuous
dual Hahn
Wilson
Laguerre Charlier
Hermite
PseudoJacobi
Bessel
‡ taken from: R. Koekoek, P. A. Lesky, and R. F. Swarttouw, Hypergeometric orthogonal polynomials and their
q-analogues, Springer-Verlag, Berlin, 2010.
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Askey scheme
(−1)-Hypergeometric orthogonal polynomials
Bannai-Ito
Big -1 Jacobi
ComplementaryBannai-Ito
ChiharaDual-1 Hahn
Little -1 Jacobi
GeneralizedHermite
[Caution] may be incomplete or wrong!
RemarkThe polynomials with q = −1 have recently been actively studied byAlexei Zhedanov and others.
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Constructions (univariate, non-terminating)
Suppose X is P - & Q-polynomial.Γ = (X,R1) : distance-regular with valency k = θ0
µΓ : the (normalized) spectral distribution of Γ on R:
µΓ
({θi√k
})=
mi
|X|(i = 0, 1, . . . , D).
Hora (1998) described the limit distributions
µΓ → µ∞ (weakly)
for various growing families of classical examples, includingHamming schemes and Johnson schemes.
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Example (Hamming schemes H(D, q))q/D → q′ ∈ (0,∞)
-2 2 4 6 8 10
0.1
0.2
0.3
0.4
0.5
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Example (Hamming schemes H(D, q))q/D → q′ ∈ (0,∞)
-2 2 4 6 8 10
0.1
0.2
0.3
0.4
0.5
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Example (Hamming schemes H(D, q))q/D → q′ ∈ (0,∞)
-2 2 4 6 8 10
0.1
0.2
0.3
0.4
0.5
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Example (Hamming schemes H(D, q))q/D → q′ ∈ (0,∞)
-2 2 4 6 8 10
0.1
0.2
0.3
0.4
0.5
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Example (Hamming schemes H(D, q))q/D → 0
-4 -2 2 4 6 8
0.1
0.2
0.3
0.4
0.5
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Example (Hamming schemes H(D, q))q/D → 0
-4 -2 2 4 6 8
0.1
0.2
0.3
0.4
0.5
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Example (Hamming schemes H(D, q))q/D → 0
-4 -2 2 4 6 8
0.1
0.2
0.3
0.4
0.5
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Example (Hamming schemes H(D, q))q/D → 0
-4 -2 2 4 6 8
0.1
0.2
0.3
0.4
0.5
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Example (Hamming schemes H(D, q))q/D → q′ ∈ [0,∞)
distribution polynomialsoriginal binomial Krawtchoukq′ > 0 Poisson Charlierq′ = 0 Gaussian Hermite
The orthogonality of the limit polynomials is almost automatic.
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Example (Johnson schemes J(N,D))N/D → N ′ ∈ [2,∞)
distribution polynomialsoriginal (no name?) dual HahnN ′ > 2 geometric MeixnerN ′ = 2 exponential Laguerre
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Hora (1998) obtained these results by evaluating the spectradirectly, which is usually quite complicated.Hora, Obata and others then revisited the results from theviewpoint of the quantum decomposition
A1 = L+ F +R
in the Terwilliger algebra of X.Their motivation is on finding concrete combinatorial models forquantum probability theory.‡
‡ See: A. Hora and N. Obata, Quantum probability and spectral analysis of graphs, Springer, Berlin, 2007.
RemarkKoornwinder (2011) found a new limit from q-Racah to big q-Jacobi inwhich the orthogonality property remains present.
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Constructions (multivariate, terminating)
X : arbitraryA = ⟨A0, . . . , AD⟩ = ⟨E0, . . . , ED⟩ : the Bose–Mesner algebra
Observation (Delsarte, 1973)Symn(A) : the BM algebra of an association scheme (∀n ∈ N)
extension of length n
ExampleD = 1 =⇒ H(n, |X|)
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Theorem (Mizukawa–T. (2004))The eigenmatrices of Symn(A) are described by the D-variateKrawtchouk polynomials.
Example (D = 2)The bivariate Krawtchouk polynomials in the variables x, y aregiven in terms of the Aomoto–Gelfand hypergeometric series:∑
a,b,c,d∈Z⩾0
a+b+c+d⩽n
(−i)a+b(−j)c+d(−x)a+c(−y)b+d
(−n)a+b+c+d a! b! c! d!αaβbγcδd
for i, j ∈ Z⩾0 with i+ j ⩽ n, for some fixed α, β, γ, δ ∈ R.
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RemarkThere are some other examples which give rise to multivariateorthogonal polynomials, e.g., the non-binary Johnson schemes,but these examples are imprimitive.
defined by Tarnanen–Aaltonen–Goethals (1985),but studied earlier by Dunkl (1976)
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Constructions (multivariate, non-terminating)
Suppose D = 2 =⇒ X : P - & Q-polynomialP is of the form
P =
1 k k1 r −r − 11 s −s− 1
,
where |X| = 1 + k + k and k ⩾ r ⩾ 0 > s.Γ = (X,R1), Γ = (X,R2) : strongly regular
complement of Γ
RemarkWhen D = 1,
Γ = (X,R1) = K|X| (complete graph).
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The adjacency matrices in Symn(A) corresponding tothe degree 1 Krawtchouk polynomials are
A =n∑
ℓ=1
I ⊗ · · · ⊗ I ⊗A1⌢ℓ
⊗ I ⊗ · · · ⊗ I,
A =
n∑ℓ=1
I ⊗ · · · ⊗ I ⊗A2⌢ℓ
⊗ I ⊗ · · · ⊗ I.
A (resp. A) : the adjacency matrix of Γn (resp. Γn)
nth Cartesian power of Γ
RemarkWhen D = 1, Γn = (K|X|)
n is the Hamming graph H(n, |X|).
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The eigenspaces of Symn(A) are parameterized by
{(i, j) : i, j ∈ Z⩾0, i+ j ⩽ n}.
Consider the joint spectrum of Γn,Γn:((θ0,0, θ0,0)
m0,0
· · ·· · ·
(θi,j , θi,j)
mi,j
· · ·· · ·
)µΓn,Γn : the (normalized) joint spectral distribution of Γn,Γn on R2:
µΓn,Γn
({(θi,j√k,θi,j√h
)})=
mi,j
|X|n(i, j ∈ Z⩾0, i+ j ⩽ n),
where k = θ0,0 (resp. h = θ0,0) is the valency of Γn (resp. Γn).
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valency of Γ valency of ΓLetk/n→ k′, k/n→ k′, r/n→ r′, s/n→ s′.
[Note] Γ is not fixed and may vary with n.
Theorem (Morales–Obata–T.)We have r′ = 0 or s′ = 0, and
µΓn,Γn → µ∞ (weakly)
where µ∞ takes one of the following:1 k′ > 0, k′ = −s′ > 0, r′ = 0: bivariate Poisson;2 k′ = r′ > 0, k′ > 0, s′ = 0: bivariate Poisson;3 k′ + k′ > 0, r′ = s′ = 0: product of Poisson and Gaussian;4 k′ = k′ = r′ = s′ = 0: bivariate Gaussian
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Example (Paley graphs Paley(q))Γ = Paley(q) (q ≡ 1 mod 4)
q/n→ q′ ∈ (0,∞)
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Example (Paley graphs Paley(q))Γ = Paley(q) (q ≡ 1 mod 4)
q/n→ q′ ∈ (0,∞)
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Example (Paley graphs Paley(q))Γ = Paley(q) (q ≡ 1 mod 4)
q/n→ q′ ∈ (0,∞)
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Example (Paley graphs Paley(q))Γ = Paley(q) (q ≡ 1 mod 4)
q/n→ q′ ∈ (0,∞)
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Example (Paley graphs Paley(q))Γ = Paley(q) (q ≡ 1 mod 4)
q/n→ 0
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Example (Paley graphs Paley(q))Γ = Paley(q) (q ≡ 1 mod 4)
q/n→ 0
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Example (Paley graphs Paley(q))Γ = Paley(q) (q ≡ 1 mod 4)
q/n→ 0
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Example (Paley graphs Paley(q))Γ = Paley(q) (q ≡ 1 mod 4)
q/n→ 0
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distribution polynomialsoriginal trinomial bivariate Krawtchouk
1 , 2 bivariate Poisson bivariate Charlier3 Poisson × Gaussian bivariate Charlier–Hermite4 bivariate Gaussian bivariate Hermite
seems previously unnoticed
1 k′ > 0, k′ = −s′ > 0, r′ = 0
2 k′ = r′ > 0, k′ > 0, s′ = 0
3 k′ + k′ > 0, r′ = s′ = 0
4 k′ = k′ = r′ = s′ = 0
Again, our motivation comes from quantum probability theory.
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Constructions (non-symmetric, terminating)
There is a fundamental connection:AW polynomials ←→ the DAHA of type (C∨
1 , C1)
Fix κ0, κ1, κ′0, κ
′1 ∈ C\{0}.
H = H(κ0, κ1, κ′0, κ′1; q) : the DAHA of type (C∨1 , C1)
generators: T ±10 , T ±1
1 ,X±1
relations: (Ti − κi)(Ti + κ−1i ) = 0 (i = 0, 1)
(T ′i − κ′i)(T ′
i + κ′−1i ) = 0 (i = 0, 1)
whereT ′0 := q−1/2XT −1
0 , T ′1 := X−1T −1
1 .
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f(z) ∈ C[z, z−1] : a Laurent polynomial
f : symmetric def⇐⇒ f(z) = f(z−1)
f : non-symmetric def⇐⇒ f(z) ̸= f(z−1)
{symmetric Laurent polynomials in z} = {polynomials in z + z−1}
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Recall the AW polynomials
fj(x) = 4ϕ3
(q−j , s∗qj+1, q−y, sqy+1
r1q, r2q, r3q
∣∣∣∣ q; q) (j = 0, 1, 2, . . . ),
which we view now as symmetric Laurent polynomials in thevariable z = s1/2q1/2+y :
fj(z) = 4ϕ3
(q−j , s∗qj+1, (sq)1/2z−1, (sq)1/2z
r1q, r2q, r3q
∣∣∣∣ q; q)
There is a faithful irreducible representation
π : H → End(C[z, z−1])
such that the symmetric AW polynomials are eigenfunctions of
X + X−1.
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f̃j(z) : the AW polynomials with parameters s̃, s̃∗, r̃1, r̃2, r̃3, where
s̃ = sq2, s̃∗ = s∗q2, r̃1 = r1q, r̃2 = r2q, r̃3 = r3q2.
The non-symmetric AW polynomials
e±j(z) ∈ C[z, z−1] (j = 0, 1, 2, . . . )
are eigenfunctions ofX±1,
and are written as C-linear combinations of
fj(z),(1− (sq)1/2z)(1− r3(q/s)
1/2z)
zf̃j−1(z).
symmetricnon-symmetric
symmetric
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Suppose X is P - & Q-polynomial of q-Racah typeSuppose further that Γ = (X,R1) has a clique C of size
|C| = 1− k
θD(Hoffmann bound).
must be the least eigenvalue of ΓFix γ ∈ C.
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The distance partition from γ : equitable
Γ0(γ) Γ1(γ) Γ2(γ) Γ3(γ) Γ4(γ)
Γi(γ) = {δ ∈ X : ∂Γ(γ, δ) = i}r3 = q−D−1
This affords the regular representation of A, and captures the(symmetric) q-Racah polynomials fj(z).[Keywords] primary T (γ)-module, Leonard pairs
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The distance partition from C : equitable
C0
C1
C2
C3
Ci = {δ ∈ X : ∂Γ(C, δ) = i}
There is associated another set of q-Racah polynomials.
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The distance partition from both γ and C : equitable
Γ0(γ) Γ1(γ) Γ2(γ) Γ3(γ) Γ4(γ)
C0
C1
C2
C3
Lee (2013, 2017) defined an irreducible H-module structure here,and then gave a combinatorial proof of the orthogonality of thenon-symmetric q-Racah polynomials.
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Lee and T. (2017+) applied this approach to the dual polarschemes, and obtained the non-symmetric dual q-Krawtchoukpolynomials.We encounter nil-DAHAs defined by Cherednik and Orr (2015).For (C∨
1 , C1), the definition reads as follows:
Fix κ, κ′ ∈ C\{0}.There is a flexibility in the definition.
H = H(κ, κ′) : a nil-DAHA of type (C∨1 , C1)
generators: T ±1,U ,X±1
relations: (T − κ)(T + κ−1) = U(U + 1) = 0
(T ′ − κ′)(T ′ + κ′−1) = U ′2 = 0
whereT ′ = XT −1, U ′ = X−1(U + 1).
Hajime Tanaka Orthogonal polynomials Novemver 25, 2017 33 / 33