Association schemes on measure spaces...Association schemes on measure spaces Alexander Barg University of Maryland June 3, 2014 Joint work with Maxim Skriganov, St. Petersburg A

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Association schemes on measure spaces

Alexander Barg

University of Maryland

June 3, 2014

Joint work with Maxim Skriganov, St. Petersburg

A. Barg (University of Maryland) Association schemes on measure spaces June 3, 2014 1 / 19

Introduction

Association schemes

Let X be a set, R = ∪i≥0Ri a partition of X × X

(X ,R) form a scheme if R0 = diag; for all i , Rti ∈ R

For (x , y) ∈ Rk , let pij(x , y) = |{z : (x , z) ∈ Ri , (z, y) ∈ Rj}|. Then

pij(x , y) = pkij

(P.-H. Zieschang, 2005)

Problem: Duality theory does not generalize(consider X a countable discrete Abelian group)


Introduction

Association schemes

Let X be a set, R = ∪i≥0Ri a partition of X × X

(X ,R) form a scheme if R0 = diag; for all i , Rti ∈ R

For (x , y) ∈ Rk , let pij(x , y) = |{z : (x , z) ∈ Ri , (z, y) ∈ Rj}|. Then

pij(x , y) = pkij

(P.-H. Zieschang, 2005)

Problem: Duality theory does not generalize(consider X a countable discrete Abelian group)


General Association Schemes: First Properties

General association schemes

Definition: X an arbitrary set with a σ-additive measure µ;R = {Ri ⊂ X × X ; i ∈ Υ} measurable sets; Υ a finite or countably infinite set.

(i) For any i ∈ Υ and any x ∈ X the set {y ∈ X : (x , y) ∈ Ri} is measurable,and its measure is finite.

(ii) R0 := {(x , x) : x ∈ X} ∈ R(iii) X × X = ∪i∈ΥRi , Ri ∩ Rj = ∅ if i = j

(iv) tRi = Ri′ , where i ′ ∈ Υ and tRi = {(y , x) | (x , y) ∈ Ri}.(v) For any i , j ∈ Υ and x , y ∈ X let

pij(x , y) = µ({z ∈ X : (x , z) ∈ Ri , (z, y) ∈ Rj}

).

For any (x , y) ∈ Rk , k ∈ Υ, the quantities pij(x , y) = pkij are constants that

depend only on k . Moreover, pkij = pk

ji .

Call X = (X , µ,R) a commutative association scheme



First properties

µi = p0ii′ := µ({y ∈ X : (x , y) ∈ Ri}), i ∈ Υ

µi = µi′ ,∑i∈Υ

µi = µ(X )

µ0 = µ({x})�

�

�

�

Classification of association schemes X = (X , µ,R) :

(S1) µ(X ) <∞ and µ0 > 0 (classical case)

(S2) µ(X ) = ∞ and µ0 > 0. In this case X is a scheme on a countablediscrete set.

(S3) µ(X ) <∞ and µ0 = 0. Their duals are schemes of type (S2).

(S4) µ(X ) = ∞ and µ0 = 0. Similarly to (S1), a scheme can be self-dual.



Adjacency algebras

X = (X , µ,R)

Let ai(x , y) = 1{(x , y) ∈ Ri}, i ∈ Υ

∑i∈Υ

ai(x , y) = j(x , y), x , y ∈ X

(a ∗ b)(x , y) =∫

Xa(x , z)b(z, y)dµ(z)

(ai ∗ aj)(x , y) =∑k∈Υ

pkij ak (x , y)

µi =

∫X

ai(x , y)dµ(y) =∫

Xai(x , y)dµ(x)

A(X ) = {a(x , y) =∑

i∈Υ ciai(x , y)} - finite linear combinations



Adjacency algebras

X = (X , µ,R)

Let ai(x , y) = 1{(x , y) ∈ Ri}, i ∈ Υ

∑i∈Υ

ai(x , y) = j(x , y), x , y ∈ X

(a ∗ b)(x , y) =∫

Xa(x , z)b(z, y)dµ(z)

(ai ∗ aj)(x , y) =∑k∈Υ

pkij ak (x , y)

µi =

∫X

ai(x , y)dµ(y) =∫

Xai(x , y)dµ(x)

A(X ) = {a(x , y) =∑

i∈Υ ciai(x , y)} - finite linear combinations



Adjacency algebras: Operator realization

Ai f (x) :=∫

Xai(x , y)f (y)dµ(y), f ∈ L2(X , µ)

Adjacency algebra:

Af (x) =∫

Xa(x , y)f (y)dµ(y), a(x , y) ∈ A(X )

.Lemma..

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The operators A are bounded, normal. Let Υ0 = {i ∈ Υ : µi > 0}. Letµ(X ) <∞, then ∑

i∈Υ0

Ai = J = µ(X )P

AiAj =∑

k∈Υ0

pkij Ak

where both the series converge in the operator norm.



Adjacency algebras: Operator realization

Ai f (x) :=∫

Xai(x , y)f (y)dµ(y), f ∈ L2(X , µ)

Adjacency algebra:

Af (x) =∫

Xa(x , y)f (y)dµ(y), a(x , y) ∈ A(X )

.Lemma..

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The operators A are bounded, normal. Let Υ0 = {i ∈ Υ : µi > 0}. Letµ(X ) <∞, then ∑

i∈Υ0

Ai = J = µ(X )P

AiAj =∑

k∈Υ0

pkij Ak

where both the series converge in the operator norm.



Main problem

Let Ei , i ∈ Υ1 be the projectors on the common eigenspaces Vj ,dimVj = mj .

Ai =∑j∈Υ1

pi(j)Ei , i ∈ Υ0∑j∈Υ1

mj |pi(j)|2 = µ(X )µi

Under which conditions are Ei contained in the adjacency algebra?



Main problem

Let Ei , i ∈ Υ1 be the projectors on the common eigenspaces Vj ,dimVj = mj .

Ai =∑j∈Υ1

pi(j)Ei , i ∈ Υ0∑j∈Υ1

mj |pi(j)|2 = µ(X )µi

Under which conditions are Ei contained in the adjacency algebra?


Schemes on Topological Groups and Spectrally Dual Partitions

Topological Abelian groups

X a topological group, X its dual group

ϕ(x + y) = ϕ(x)ϕ(y), (ϕ · ψ)(x) = ϕ(x)ψ(x)

ϕ(x) = ϕ(−x).; x , y ∈ X ;ϕ, ψ ∈ X

Fourier transforms:

F∼ : f (x) → f (ξ) =∫

Xξ(x)f (x)dµ(x), ξ ∈ X

F ♮ : g(ξ) → g♮(x) =∫

Xξ(x)g(ξ)d µ(ξ), x ∈ X .

F∼F ♮ = I, F ♮F∼ = I



Topological Abelian groups

X a topological group, X its dual group

ϕ(x + y) = ϕ(x)ϕ(y), (ϕ · ψ)(x) = ϕ(x)ψ(x)

ϕ(x) = ϕ(−x).; x , y ∈ X ;ϕ, ψ ∈ X

Fourier transforms:

F∼ : f (x) → f (ξ) =∫

Xξ(x)f (x)dµ(x), ξ ∈ X

F ♮ : g(ξ) → g♮(x) =∫

Xξ(x)g(ξ)d µ(ξ), x ∈ X .

F∼F ♮ = I, F ♮F∼ = I



Spectrally dual partitions

N = {Ni , i ∈ Υ}, Ni = {x ∈ X : (x , 0) ∈ Ri}

N = {Ni , i ∈ Υ}, Ni = {ϕ ∈ X : (ϕ, 1) ∈ Ri}

Let Λ2(N ) ∈ L2(X , µ) be the space of functions piecewise constant on thepartition N . Similarly define Λ2(N )..Definition..

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Let X , X be mutually dual Abelian groups. The partitions N and N are calledspectrally dual if

F∼Λ2(N ) = Λ2(N ), Λ2(N ) = F ♮Λ2(N )



Overview of the main results

(i) Let N , N be spectrally dual partitions of X , X . Then X (X , µ,R) andX (X , µ, R) form translation invariant association schemes.

(ii) If at least one of X , X is connected, then the pair (X , X ) does not supportspectrally dual partitions.

(iii) Let X be totally disconnected (0-dimensional) LCA group, X its dual group(e.g., Cantor-type groups, groups of p-adic numbers). We construct dualpairs of association schemes, describe their adjancency algebras, andcompute their eigenvalues and intersection numbers. The schemes aremetric nonpolynomial in accordance with the metric on X .

(iv) For (X , X ) as in (iii) we also construct nonmetric schemes and computetheir eigenvalues. We observe that the eigenvalues coincide withevaluations of wavelet-like functions on X and establish basic properties ofthese wavelets.


Schemes on topological Abelian groups

Schemes on Abelian groups

Let N , N be a pair of spectrally dual partitions of X , X . Define

χi(x , y) = χi(x − y), χi(ϕψ−1) = χi(ϕψ

−1)

Ai f (x) =∫

Xχi(x − y)f (y)dy , Aig(ϕ) =

∫Xχi(ϕψ

−1)g(ψ)dψ

Ek f (x) =∫

Xχ♮

k (x − y)f (y)dy , Ek g(ϕ) =∫

Xχk (ϕψ

−1)g(ψ)dψ

Then

Ei =∑

k∈Υ0

qi(k)Ak , i ∈ Υ0

Aj =∑

k∈Υ0

pj(k)Ek , j ∈ Υ0.


Schemes on Totally Disconnected groups

Totally disconnected groups

A topological Abelian group X is called totally disconnected if its connectedcomponents are points. Note that X is zero-dimensional.The topology on X is defined by the chain

X = X0 ⊃ X1 ⊃ · · · ⊃ Xj ⊃ · · · ⊃ {0}

Xj are subgroups of finite index ≥ 2; ∩j≥0Xj = {0}.

Examples: Cantor-type groups (e.g., {0,1}ω); p-adic integers.

The topology is metrizable, by a non-Archimedean valuationν(x) = max{j : x ∈ Xj}, x = 0. The subgroups Xj form a sequence of nestedballs:

Xj = {x ∈ X : ρ(x) ≤ 2−j}, j = 0, 1, . . . .

X can be identified with the set of all infinite sequences

x = (z1, z2, . . . ), zi ∈ Xi−1/Xi , i ≥ 1.





X = X0 ⊃ X1 ⊃ · · · ⊃ Xj ⊃ · · · ⊃ {0}




Xj = {x ∈ X : ρ(x) ≤ 2−j}, j = 0, 1, . . . .


x = (z1, z2, . . . ), zi ∈ Xi−1/Xi , i ≥ 1.





X = X0 ⊃ X1 ⊃ · · · ⊃ Xj ⊃ · · · ⊃ {0}




Xj = {x ∈ X : ρ(x) ≤ 2−j}, j = 0, 1, . . . .


x = (z1, z2, . . . ), zi ∈ Xi−1/Xi , i ≥ 1.



Dual groups

The character group of X is easily described.compact ⇔ discrete totally disconnected ⇔ periodic

Annihilator of Xj ⊂ X : a closed subgroup of X defined as

X⊥j := {ϕ ∈ X : ϕ(x) = 1 for all x ∈ Xj}

X⊥j

∼= (X/Xj)∧

|X⊥j | = |X/Xj |

We obtain (in the compact case):

{1} = X⊥0 ⊂ X⊥

1 ⊂ · · · ⊂ X⊥j ⊂ · · · ⊂ X , ∪j≥0X⊥

j = X .

X is countable, discrete, and periodic (every element has a finite order)



Dual groups

The character group of X is easily described.compact ⇔ discrete totally disconnected ⇔ periodic

Annihilator of Xj ⊂ X : a closed subgroup of X defined as

X⊥j := {ϕ ∈ X : ϕ(x) = 1 for all x ∈ Xj}

X⊥j

∼= (X/Xj)∧

|X⊥j | = |X/Xj |

We obtain (in the compact case):

{1} = X⊥0 ⊂ X⊥

1 ⊂ · · · ⊂ X⊥j ⊂ · · · ⊂ X , ∪j≥0X⊥

j = X .

X is countable, discrete, and periodic (every element has a finite order)



Balls and spheres

Let us number the balls by their radii:

B(r) = {x : ρ(x) ≤ r}, r ∈ Υ, B(t) = {ξ : ρ(ξ) ≤ t}, t ∈ Υ

Υ0 = {r ∈ Υ : µ(B(r)) > 0}, Υ0 = {t ∈ Υ : µ(B(t)) > 0}.

Let τ−(λ) = max{r : r < λ}, τ+(λ) = min{r : r > λ}

Spheres:

S(r) = {x ∈ X : ρ(x) = r}. r ∈ Υ,

S(t) = {ξ ∈ X : ρ(ξ) = t}, t ∈ Υ

S(0) = B(0), S(0) = B(0)

S(r) = B(r)\B(τ−(r)), S(t) = B(t)\B(τ−(t))



Balls and spheres

Let us number the balls by their radii:

B(r) = {x : ρ(x) ≤ r}, r ∈ Υ, B(t) = {ξ : ρ(ξ) ≤ t}, t ∈ Υ

Υ0 = {r ∈ Υ : µ(B(r)) > 0}, Υ0 = {t ∈ Υ : µ(B(t)) > 0}.

Let τ−(λ) = max{r : r < λ}, τ+(λ) = min{r : r > λ}

Spheres:

S(r) = {x ∈ X : ρ(x) = r}. r ∈ Υ,

S(t) = {ξ ∈ X : ρ(ξ) = t}, t ∈ Υ

S(0) = B(0), S(0) = B(0)

S(r) = B(r)\B(τ−(r)), S(t) = B(t)\B(τ−(t))



Spectral parameters (Eigenvalues).Theorem..

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Partitions of the groups X and X into spheres are symmetric spectrally dual.We have

χ(S(r), ξ) =∑b∈Υ

pr (b)χ(S(b), ξ) (1)

χ♮(S(t), x) =∑a∈Υ

qt(a)χ(S(a), ξ) (2)

where

pr (b) =

0 if b > τ+(r),−µ(B(τ−(r))) if b = τ+(r),µ(B(r))− µ(B(τ−(r))) if 0 ≤ b ≤ r

qr (a) =

0 if a > τ+(t♮),−µ(B(τ−(t))) if a = τ+(t♮)µ(B(t))− µ(B(τ−(t))) if 0 ≤ a ≤ t♮.

Relations (1) and (2) hold pointwise for all ξ ∈ X and x ∈ X.



Dual pairs of association schemes

We obtain a class of dual pairs (X , X ) of association schemes.

The intersection numbers of X are:

pr3r1,r2 =

0 if λ = 1

µ(S(r∗)) if λ = 2[|B(r∗)/B(τ−(r∗))| − 2

]µ(B(τ−(r∗))

)if λ = 3

where ri ∈ Υ, i = 1, 2, 3; r∗ := min(r1, r2, r3), and λ denotes the number oftimes max(r1, r2, r3) appears among {r1, r2, r3}.

The intersection numbers of X are given by

pt3t1,t2 =

0 if λ = 1

µ(S(t∗)) if λ = 2[|B(t∗)/B(τ−(t∗))| − 2

]µ(B(τ−(t∗))

)if λ = 3

where ti ∈ Υ, i = 1, 2, 3; t∗ and λ are defined analogously






pr3r1,r2 =

0 if λ = 1

µ(S(r∗)) if λ = 2[|B(r∗)/B(τ−(r∗))| − 2

]µ(B(τ−(r∗))

)if λ = 3



pt3t1,t2 =

0 if λ = 1

µ(S(t∗)) if λ = 2[|B(t∗)/B(τ−(t∗))| − 2

]µ(B(τ−(t∗))

)if λ = 3







pr3r1,r2 =

0 if λ = 1

µ(S(r∗)) if λ = 2[|B(r∗)/B(τ−(r∗))| − 2

]µ(B(τ−(r∗))

)if λ = 3



pt3t1,t2 =

0 if λ = 1

µ(S(t∗)) if λ = 2[|B(t∗)/B(τ−(t∗))| − 2

]µ(B(τ−(t∗))

)if λ = 3




Adjacency algebras

Letα0(x) = 1; αl(x) = χ[S(rl); x ], l ≥ 1,

A(sph) = {f : X → C | f (x) = c0α0(x) +∑l∈N

clαl(x)},

where only finitely many coefficients c0, cl are nonzero. Dually, let

βk (ξ) = χ[S(tk ); ξ], k ≥ 1

A (sph) = {g : X → C | c′0β0(ξ) +

∑k∈N

c′kβk (ξ)}

Lemma: F∼A(sph) = A (sph);F ♮A (sph) = A(sph)



Nonmetric schemes

Let X , X be a pair of LCA groups.

X ⊃ · · · ⊃ Xj−1 ⊃ Xj ⊃ · · · ⊃ {0}; ∪j∈ZXj = X , ∩j∈ZXj = {0}

WriteX =

∪r∈R

∪1≤i≤n(r)−1

Φi(r)

where Φi(r) = B(τ−(r)) + zi,r , zi,r ∈ B(r)/B(τ−(r)), 0 ≤ i ≤ n(r)− 1

We construct a pair of dual association schemes whose classes are indexedby pairs (r , j), r ∈ R, j ∈ Z. Eigenfunctions of these schemes coincide withevaluations of Haar-like wavelets on X , X .



Nonmetric schemes

Let X , X be a pair of LCA groups.

X ⊃ · · · ⊃ Xj−1 ⊃ Xj ⊃ · · · ⊃ {0}; ∪j∈ZXj = X , ∩j∈ZXj = {0}

WriteX =

∪r∈R

∪1≤i≤n(r)−1

Φi(r)

where Φi(r) = B(τ−(r)) + zi,r , zi,r ∈ B(r)/B(τ−(r)), 0 ≤ i ≤ n(r)− 1

We construct a pair of dual association schemes whose classes are indexedby pairs (r , j), r ∈ R, j ∈ Z. Eigenfunctions of these schemes coincide withevaluations of Haar-like wavelets on X , X .



Preprint:

A. Barg and M. Skriganov, Association schemes on general measure spacesand zero-dimensional Abelian groups, arXiv.org/1310.5359


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Association schemes on measure spaces...Association schemes on measure spaces Alexander Barg University of Maryland June 3, 2014 Joint work with Maxim Skriganov, St. Petersburg A