Some remarks on association schemesusers.wpi.edu/~martin/MEETINGS/LINESTALKS/Barg.pdf · Some...
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Some remarks on association schemes
Alexander BargUniversity of Maryland
A. Barg (UMD) Association schemes 1 / 29
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Association schemes
Regularity conditions in graphs
Petersen graph
N = 10 vertices, regular of degree 3
A. Barg (UMD) Association schemes 2 / 29
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Association schemes
Regularity conditions in graphs
Petersen graph
N = 10 vertices, regular of degree 3Adjacent vertices have no common neighbors
A. Barg (UMD) Association schemes 3 / 29
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Association schemes
Regularity conditions in graphs
Petersen graph
N = 10 vertices, regular of degree 3Adjacent vertices have no common neighbors
A. Barg (UMD) Association schemes 4 / 29
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Association schemes
Regularity conditions in graphs
Petersen graph
N = 10 vertices, regular of degree 3Adjacent vertices have no common neighbors
A. Barg (UMD) Association schemes 5 / 29
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Association schemes
Regularity conditions in graphs
Petersen graph
N = 10 vertices, regular of degree 3Adjacent vertices have no common neighbors
Non-adj. vertices have one common neighbor
A. Barg (UMD) Association schemes 6 / 29
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Association schemes
Regularity conditions in graphs
Petersen graph
N = 10 vertices, regular of degree 3Adjacent vertices have no common neighbors
Non-adj. vertices have one common neighbor
A. Barg (UMD) Association schemes 7 / 29
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Association schemes
Regularity conditions in graphs
Petersen graph
N = 10 vertices, regular of degree 3Adjacent vertices have no common neighbors
Non-adj. vertices have one common neighbor
Strongly regular graph (10,3,0,1)
C. Godsil, D. Royle: Alg. graph theory (2001)
(GTM 207)
A. Barg (UMD) Association schemes 8 / 29
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Association schemes
Regularity conditions in graphs
Hamming graph H3
Distance 1
Distance 3Distance 2
A. Barg (UMD) Association schemes 9 / 29
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Association schemes
Regularity conditions in graphs
Hamming graph H3
Distance 1
Distance 3Distance 2
Count of triangles
The number of triangles with a fixedbase and sides of some fixed colorsdepends only on the color of the base
A. Barg (UMD) Association schemes 10 / 29
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Association schemes
Regularity conditions in graphs
Hamming graph H3
Distance 1
Distance 3Distance 2
Count of triangles
The number of triangles with a fixedbase and sides of some fixed colorsdepends only on the color of the base
A. Barg (UMD) Association schemes 11 / 29
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Association schemes
Regularity conditions in graphs
Hamming graph H3
Distance 1
Distance 3Distance 2
Count of triangles
The number of triangles with a fixedbase and sides of some fixed colorsdepends only on the color of the base
A. Barg (UMD) Association schemes 12 / 29
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Association schemes
Regularity conditions in graphs
Hamming graph H3
Distance 1
Distance 3Distance 2
Count of triangles
The number of triangles with a fixedbase and sides of some fixed colorsdepends only on the color of the base
A. Barg (UMD) Association schemes 13 / 29
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Association schemes
Regularity conditions in graphs
Hamming graph H3
Distance 1
Distance 3Distance 2
Count of triangles
The number of triangles with a fixedbase and sides of some fixed colorsdepends only on the color of the base
A. Barg (UMD) Association schemes 14 / 29
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Association schemes
Regularity conditions in graphs
Hamming graph H3
Distance 1
Distance 3Distance 2
Count of triangles
The number of triangles with a fixedbase and sides of some fixed colorsdepends only on the color of the base
Call these numbers the intersectionnumbers of the graph
pki,j : Base of length k , sides of lengths
i , j
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Association schemes
From graphs to matrices....
Adjacency matrices:
A0 = I8 A1 =
0111000010001100100010101000011001100001010100010011000100001110
A2 =
0000111000110001010100010110000110000110100010101000110001110000
A3 =
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
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Association schemes
...to harmonic analysis
The algebra A has a basis of primitive idempotents Ei , i = 0,1,2,3which satisfy similar properties:∑
i
Ei = I, EiEj = δijEi , E0 =1|X |
J, Ei ◦ Ej =1|X |
∑k
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.
A. Barg (UMD) Association schemes 17 / 29
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Association schemes
...to harmonic analysis
The algebra A has a basis of primitive idempotents Ei , i = 0,1,2,3which satisfy similar properties:∑
i
Ei = I, EiEj = δijEi , E0 =1|X |
J, Ei ◦ Ej =1|X |
∑k
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.
A. Barg (UMD) Association schemes 17 / 29
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Association schemes
...to harmonic analysis
The algebra A has a basis of primitive idempotents Ei , i = 0,1,2,3which satisfy similar properties:∑
i
Ei = I, EiEj = δijEi , E0 =1|X |
J, Ei ◦ Ej =1|X |
∑k
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.
A. Barg (UMD) Association schemes 17 / 29
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Association schemes
...to harmonic analysis
The algebra A has a basis of primitive idempotents Ei , i = 0,1,2,3which satisfy similar properties:∑
i
Ei = I, EiEj = δijEi , E0 =1|X |
J, Ei ◦ Ej =1|X |
∑k
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.A. Barg (UMD) Association schemes 17 / 29
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Association schemes
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
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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}
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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
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Association schemes
Eigenvalues of schemes
We have
Aj =d∑
i=0
PijEi ; Ej =1|X |
d∑i=0
QijAi
In the polynomial case, the eigenvalues Pij ,Qij coincide with values ofsome discrete orthogonal polynomials
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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
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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
A. Barg (UMD) Association schemes 21 / 29
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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
A. Barg (UMD) Association schemes 21 / 29
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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
A. Barg (UMD) Association schemes 21 / 29
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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
A. Barg (UMD) Association schemes 21 / 29
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Association schemes
Bounds on codes
This gives rise to a Linear Programming problem. Letδ := min{i > 0 : ai ̸= 0}, then
M < max{ d∑
i=δ
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 )
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Association schemes
Bounds on codes
This gives rise to a Linear Programming problem. Letδ := min{i > 0 : ai ̸= 0}, then
M < max{ d∑
i=δ
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 )
A. Barg (UMD) Association schemes 22 / 29
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Association schemes
Bounds on codes
This gives rise to a Linear Programming problem. Letδ := min{i > 0 : ai ̸= 0}, then
M < max{ d∑
i=δ
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 )
A. Barg (UMD) Association schemes 22 / 29
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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}
(1)
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.
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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}
(1)
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.
A. Barg (UMD) Association schemes 23 / 29
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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}
(1)
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.
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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
pij(x , y) = pkij ; pk
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
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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
pij(x , y) = pkij ; pk
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
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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
pij(x , y) = pkij ; pk
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
A. Barg (UMD) Association schemes 24 / 29
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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⊥j
∼= (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)
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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⊥j
∼= (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)
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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⊥j
∼= (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)
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Association schemes
Metric structure of the group
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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).
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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).
A. Barg (UMD) Association schemes 29 / 29
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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 groups
We 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).
A. Barg (UMD) Association schemes 29 / 29
![Page 53: Some remarks on association schemesusers.wpi.edu/~martin/MEETINGS/LINESTALKS/Barg.pdf · Some remarks on association schemes Alexander Barg University of Maryland A. Barg (UMD) Association](https://reader034.fdocuments.in/reader034/viewer/2022042310/5ed83d800fa3e705ec0e1998/html5/thumbnails/53.jpg)
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).
A. Barg (UMD) Association schemes 29 / 29
![Page 54: Some remarks on association schemesusers.wpi.edu/~martin/MEETINGS/LINESTALKS/Barg.pdf · Some remarks on association schemes Alexander Barg University of Maryland A. Barg (UMD) Association](https://reader034.fdocuments.in/reader034/viewer/2022042310/5ed83d800fa3e705ec0e1998/html5/thumbnails/54.jpg)
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).
A. Barg (UMD) Association schemes 29 / 29