Some remarks on association schemesusers.wpi.edu/~martin/MEETINGS/LINESTALKS/Barg.pdf · Some...

Some remarks on association schemes

Alexander BargUniversity of Maryland

A. Barg (UMD) Association schemes 1 / 29

Association schemes

Regularity conditions in graphs

Petersen graph

N = 10 vertices, regular of degree 3

Association schemes

Petersen graph

N = 10 vertices, regular of degree 3Adjacent vertices have no common neighbors

Association schemes

Petersen graph

Association schemes

Petersen graph

Association schemes

Petersen graph

Non-adj. vertices have one common neighbor

Association schemes

Petersen graph

Association schemes

Petersen graph

Strongly regular graph (10,3,0,1)

C. Godsil, D. Royle: Alg. graph theory (2001)

(GTM 207)

Association schemes

Hamming graph H3

Distance 1

Distance 3Distance 2

Association schemes

Hamming graph H3

Distance 1

Count of triangles

The number of triangles with a fixedbase and sides of some fixed colorsdepends only on the color of the base

Association schemes

Hamming graph H3

Distance 1

Count of triangles

Association schemes

Hamming graph H3

Distance 1

Count of triangles

Association schemes

Hamming graph H3

Distance 1

Count of triangles

Association schemes

Hamming graph H3

Distance 1

Count of triangles

Association schemes

Hamming graph H3

Distance 1

Count of triangles

Call these numbers the intersectionnumbers of the graph

pki,j : Base of length k , sides of lengths

Association schemes

From graphs to matrices....

Adjacency matrices:

A0 = I8 A1 =

0111000010001100100010101000011001100001010100010011000100001110

0000111000110001010100010110000110000110100010101000110001110000

0000000100000010000001000000100000010000001000000100000010000000

∑i≥0

Ai = J; AiAj =∑k≥0

pki,jAk = AjAi ; Ai ◦ Aj = δijAi

The matrices Ai span a 4-dim complex commutative algebra A closedwith respect to pointwise product and convolution

Association schemes

...to harmonic analysis

The algebra A has a basis of primitive idempotents Ei , i = 0,1,2,3which satisfy similar properties:∑

Ei = I, EiEj = δijEi , E0 =1|X |

J, Ei ◦ Ej =1|X |

qkij Ek

The polynomial case:

Ai = pi(A1),

where pi is a polynomial of degree i .

These polynomials satisfy a three-term recurrence of the form

xpi = aipi+1 + bipi + cipi−1

and hence form a family of orthogonal polynomials.

Association schemes

J, Ei ◦ Ej =1|X |

qkij Ek

Ai = pi(A1),

Association schemes

J, Ei ◦ Ej =1|X |

qkij Ek

Ai = pi(A1),

Association schemes

J, Ei ◦ Ej =1|X |

qkij Ek

Ai = pi(A1),

and hence form a family of orthogonal polynomials.A. Barg (UMD) Association schemes 17 / 29

Association schemes

X a finite set

Commutative Association Scheme A = (X ,R = (R0,R1, . . . ,Rd)) is apartition of X × X such that1. R0 = {(x , x), x ∈ X}2. if (x , y) ∈ Ri , ∃ i ′ ∈ {1, . . . , d} such that (y , x) ∈ Ri ′

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

pij(x , y) = pkij ; pk

ij = pkji

(Bose et al. ’52; ’59; Delsarte ’73)

• Bannai-Ito, Alg. Combin., 1981;• Brouwer-Cohen-Neumaier, Dist. Reg. Graphs, 1989• Martin-Tanaka, Eur. J. Combin., 2009

Association schemes

Abelian (translation) schemes and duality

Let X be a finite Abelian group with the structure of an associationscheme.

A scheme X is called a translation scheme if (x , y) ∈ Ri implies that(x + z, y + z) ∈ Ri for all z ∈ X .

For translation schemes, duality can be expressed in terms of thepartition of X

For translation schemes we can consider blocks

Ni := {x ∈ X : (x ,0) ∈ Ri}.

Let X̂ be the dual group of X , then we can define a dual scheme onX̂ × X̂ by setting

N̂i = {χ ∈ X̂ : Eiχ = χ}

Association schemes

Abelian (translation) schemes and duality

Let X be a finite Abelian group with the structure of an associationscheme.

A scheme X is called a translation scheme if (x , y) ∈ Ri implies that(x + z, y + z) ∈ Ri for all z ∈ X .

For translation schemes, duality can be expressed in terms of thepartition of X

For translation schemes we can consider blocks

Ni := {x ∈ X : (x ,0) ∈ Ri}.

Let X̂ be the dual group of X , then we can define a dual scheme onX̂ × X̂ by setting

N̂i = {χ ∈ X̂ : Eiχ = χ}A. Barg (UMD) Association schemes 19 / 29

Association schemes

Eigenvalues of schemes

We have

Aj =d∑

PijEi ; Ej =1|X |

d∑i=0

In the polynomial case, the eigenvalues Pij ,Qij coincide with values ofsome discrete orthogonal polynomials

Association schemes

Codes (Subsets in Schemes)

C ⊂ X , |C| = M

χ = χC := (χ1, . . . , χ|X |), where χi = 1(xi ∈ C)

ai =1M

χT Aiχ, i = 0,1, . . . , d

The MacWilliams theorem:

(a · Q)j =|X |M

(χT Ejχ)

Corollary (Delsarte’s inequalities): (aQ)j ≥ 0 for all j

Definition: C is called a t-design if (aQ)j = 0 for j = 1, . . . , t

Association schemes

C ⊂ X , |C| = M

ai =1M

χT Aiχ, i = 0,1, . . . , d

(a · Q)j =|X |M

(χT Ejχ)

Association schemes

C ⊂ X , |C| = M

ai =1M

χT Aiχ, i = 0,1, . . . , d

(a · Q)j =|X |M

(χT Ejχ)

Association schemes

C ⊂ X , |C| = M

ai =1M

χT Aiχ, i = 0,1, . . . , d

(a · Q)j =|X |M

(χT Ejχ)

Association schemes

C ⊂ X , |C| = M

ai =1M

χT Aiχ, i = 0,1, . . . , d

(a · Q)j =|X |M

(χT Ejχ)

Association schemes

Bounds on codes

This gives rise to a Linear Programming problem. Letδ := min{i > 0 : ai ̸= 0}, then

M < max{ d∑

ui s.t. ui ≥ 0,∑

uiQij ≥ 0, j = 1, . . . , d}

If the scheme is co-metric (Q-polynomial), then the Delsarte inequalities canbe written in polynomial form.

Kabatiansky-Levenshtein theory ’78: This approach extends to a large classof spaces, in particular, Sn−1(K ),K = R,C; KPn−1; Gk,n(K )

Association schemes

Bounds on codes

M < max{ d∑

ui s.t. ui ≥ 0,∑

uiQij ≥ 0, j = 1, . . . , d}

Association schemes

Bounds on codes

M < max{ d∑

ui s.t. ui ≥ 0,∑

uiQij ≥ 0, j = 1, . . . , d}

Association schemes

Bounds on codes

In particular

for Sn−1(R) we have

M(δ) ≤ max{

f (1), where f (t) = 1 +∑i≥1

fiGk (t), fi ≥ 0;

f (t) ≤ 0,−1 ≤ t ≤ δ}

Gegenbauer polynomials∫ 1−1 Gi(t)Gj(t)(1 − t2)(n−3)/2dt = δij , i ̸= j

For instance for equiangular lines t ∈ {a,−a}Spherical designs: C ⊂ Sn−1 :

∑x,y∈C Gk (⟨x , y⟩) = 0, k = 1, . . . , t

for CPn−1 we have P(n−2)/2,0k (2t2 − 1).

Taking f = 1 + f1P(n−2)/2,01 , we obtain the Welch bound.

Association schemes

Bounds on codes

In particular

M(δ) ≤ max{

f (1), where f (t) = 1 +∑i≥1

fiGk (t), fi ≥ 0;

f (t) ≤ 0,−1 ≤ t ≤ δ}

For instance for equiangular lines t ∈ {a,−a}

Spherical designs: C ⊂ Sn−1 :∑

x,y∈C Gk (⟨x , y⟩) = 0, k = 1, . . . , t

Association schemes

Bounds on codes

In particular

M(δ) ≤ max{

f (1), where f (t) = 1 +∑i≥1

fiGk (t), fi ≥ 0;

f (t) ≤ 0,−1 ≤ t ≤ δ}

For instance for equiangular lines t ∈ {a,−a}Spherical designs: C ⊂ Sn−1 :

∑x,y∈C Gk (⟨x , y⟩) = 0, k = 1, . . . , t

Association schemes

Schemes on infinite sets

Attempt to define a scheme on X using the classical definition(P.-H. Zieschang)For (x , y) ∈ Rk , let pij(x , y) = |{z : (x , z) ∈ Ri , (y , z) ∈ Rj}|. Then

ij = pkji

Consider the group case: X = Z; X̂ = S1

More generally, if X is an Abelian group, countable, discrete andperiodic then X̂ is uncountable, compact, zero-dimensional

Association schemes

ij = pkji

Association schemes

ij = pkji

Association schemes

Association schemes on infinite sets

An approach (the set is still finite):

Define a norm on Fnq as follows: ∥0∥ = 0 and for x ̸= 0

∥x∥ = i if xi ̸= 0 and xi+1 = · · · = xn = 0

Construct a metric association scheme on X = Fnq with classes

Ri = {(x , y) : ∥x − y∥ = i}, i = 0,1, . . . , n.

This scheme (together with its extension) was analyzed by Martin andStinson (’98). The eigenvalues are evaluations of n-variatepolynomials.

Association schemes

∥x∥ = i if xi ̸= 0 and xi+1 = · · · = xn = 0

Ri = {(x , y) : ∥x − y∥ = i}, i = 0,1, . . . , n.

Association schemes

∥x∥ = i if xi ̸= 0 and xi+1 = · · · = xn = 0

Ri = {(x , y) : ∥x − y∥ = i}, i = 0,1, . . . , n.

Association schemes

Totally disconnected (0-dimensional) groups

A topological Abelian group X is called totally disconnected if itsconnected components 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 ofnested balls:

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.

Association schemes

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

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

Association schemes

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

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

Association schemes

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

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

Association schemes

Dual groups

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

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

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

∼= (X/Xj)∧

|X⊥j | = |X/Xj |

We obtain (in the compact case):

{1} = X⊥0 ⊂ X⊥

1 ⊂ · · · ⊂ X⊥j ⊂ · · · ⊂ X̂ , ∪j≥0X⊥

j = X̂ .

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

Association schemes

Dual groups

∼= (X/Xj)∧

|X⊥j | = |X/Xj |

{1} = X⊥0 ⊂ X⊥

1 ⊂ · · · ⊂ X⊥j ⊂ · · · ⊂ X̂ , ∪j≥0X⊥

j = X̂ .

Association schemes

Dual groups

∼= (X/Xj)∧

|X⊥j | = |X/Xj |

{1} = X⊥0 ⊂ X⊥

1 ⊂ · · · ⊂ X⊥j ⊂ · · · ⊂ X̂ , ∪j≥0X⊥

j = X̂ .

Association schemes

Metric structure of the group

Association schemes

Association schemes on 0-dimensional groups

We explicitly compute the eigenvalues and intersection numbersof the schemes on compact and LCA groups. Eigenvalues for aclass of wavelet bases on the group.We compute dual schemes of the schemes on Abelian groupsWe describe the adjacency algebras of the constructed schemes

Work with Maxim Skriganov“Association schemes on general measure spaces and zero-dimensionalAbelian groups,” Adv. Math., vol. 281, 2015, pp. 142–247 (arXiv 1310.5359).

Association schemes

We explicitly compute the eigenvalues and intersection numbersof the schemes on compact and LCA groups. Eigenvalues for aclass of wavelet bases on the group.

We compute dual schemes of the schemes on Abelian groupsWe describe the adjacency algebras of the constructed schemes

Association schemes

We explicitly compute the eigenvalues and intersection numbersof the schemes on compact and LCA groups. Eigenvalues for aclass of wavelet bases on the group.We compute dual schemes of the schemes on Abelian groups

We describe the adjacency algebras of the constructed schemes

Association schemes

Some remarks on association schemesusers.wpi.edu/~martin/MEETINGS/LINESTALKS/Barg.pdf · Some...

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