Scaffolds A graph-based system for computations with certain tensors - - PowerPoint PPT Presentation

Examples Definitions and Basics Some Results of Dickie and Suzuki

Scaffolds

A graph-based system for computations with certain tensors William J. Martin

Department of Mathematical Sciences and Department of Computer Science Worcester Polytechnic Institute

Mini Symposium on Spectral Graph Theory and Related Topics JCCA 2018, Sendai, Japan May 23, 2018

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki

The Other Talk

I am sorry that I will give a different talk than the one promised.

◮ Examples of Q-polynomial association schemes (“DRG

duals”) [see also King, below]

◮ new hemisystems in generalized quadrangles (d = 4,

Q-antipodal, w = 2)

◮ Penttila-Williford family (2011) from relative hemisystems

(d = 4, Q-bipartite, and d = 3, primitive)

◮ Moorhouse-Williford family (2016) of double covers of

symplectic dual polar graphs using Maslov index (d odd, Q-bipartite, sometimes irrational) with two Q-polynomial

• rderings

◮ new linked systems of symmetric designs (LSSDs) by

Davis/WJM/Polhill, Jedwab/Li/Simon, Kodalen (d = 3, Q-antipodal)

◮ new sets of of real mutually unbiased bases (d = 4,

Q-antipodal & Q-bipartite)

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki

The Other Talk, continued

◮ characterization of LSSDs in terms of “linked simplices”

(Kodalen)

◮ Nearest neighbor graph (WJM/Williford):

◮ With m1 = Q01 > Q11 > · · · > Qd1, A = A1 ◮ √m1 ≤ v1 ≤ τm−1 and m1 ≤ v1 + vi whenever i > 1 with

pi

11 > 0

◮ “Semidefinite programming”: Kodalen/WJM/Yu apply

Schoenberg’s Theorem. Evaluating the Gegenbauer polynomial Gd+2(t) on E1 for a d-class Q-polynomial scheme rules out infinitely many feasible parameter sets

◮ Gavrilyuk/Vidali: integralilty of triple intersection numbers

rule out many feasible parameter sets in the Williford tables, as well as three infinite families (d = 3, 4, 5)

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki

Gavin King’s Exceptional Schurian Cometric Schemes

Group G acts multiplicity-freely on the left cosets of subgroup H

d |X| struc multiplicities valencies G , H 3 1288 P

1, 22, 230, 1035

1, 165, 330, 792

M23 , M11 4 11178 P

1, 23, 275, 2024, 8855

1, 1100, 5600, 4125, 352

Co3 , HS 2Q (02431) 4 13056 P

1, 135, 3400, 8925, 595

1, 210, 1575, 5600, 5670

Sp(8, 2) , S10 4 28431 P

1, 260, 9450, 18200, 520

1, 960, 3150, 22400, 1920

O+

8 (3).2 , O+ 8 (2).2

5 352 A

1, 21, 154, 154, 21, 1

1, 35, 105, 126, 70, 15

M22.2 , A7 5 28160 A

1, 429, 13650, 13650, 429, 1

1, 364, 3159, 12636, 10920, 1080

Fi22.2 , O7(3) 5 104448 P

1, 187, 7700, 56100, 39270, 1190

1, 462, 5775, 30800, 62370, 5040

PSO−(10, 2) , S12 6 704 AB

1, 22, 175, 308, 175, 22, 1

1, 50, 175, 252, 175, 50, 1

HS.2 , U3(5) 2Q (0523416) 7 4050 A

1, 22, 252, 1750, 1750, 252, 22, 1

1, 176, 462, 1155, 1232, 672, 330, 22

McL.2 , M22

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Introduction Exercises

The Simplest Scaffolds

All matrices here are square. Rows and columns are indexed by some finite set X.

◮ M

denotes the diagonal of matrix M

◮ M

equals the trace of M

◮ Matrix N, as a second-order tensor, is represented as N ◮ The sum of all entries of N is N ◮ The ordinary matrix product of M and N is encoded as a

series reduction

M N ◮ Entrywise multiplication is a parallel reduction M N

.

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Introduction Exercises

Example

X = {1, 2, 3}, M =   −2 3 3 3 −2 3 3 3 −2  , N =   1/3 1/3 1/3 1/3 1/3  

◮ M

=   −2 −2 −2  

M

= −6

◮ N

=   1/3 1/3 1/3 1/3 1/3  

◮ while N

= 5/3

◮ But N

=

• 1/3

2/3 2/3

• ,

N

=

• 2/3

2/3 1/3 ⊤

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Introduction Exercises

Example, continued

X = {1, 2, 3}, M =   −2 3 3 3 −2 3 3 3 −2  , N =   1/3 1/3 1/3 1/3 1/3  

◮ Products: M N

=   1 1/3 1/3 −2/3 2 1/3 1 1/3 2  

◮ while M N

=   1 1 1 1 1  

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

Definition

Let X be a finite set and let A be a vector subspace of MatX(C). Our standard basis for V = CX is {ˆ x | x ∈ X}. Given

◮ A (di)graph G = (V (G), E(G)) ◮ A subset R ⊆ V (G) of “red” nodes, and ◮ a map from edges of G to matrices in A:

w : E(G) → A (edge weights) The scaffold S(G; R, w) is defined as the quantity S(G; R, w) =

• ϕ:V (G)→X
• e∈E(G)

e=(a,b)

w(e)ϕ(a),ϕ(b)

• r∈R
• ϕ(r).

This is an element of V ⊗|R|, so we say S(G; R, w) is a scaffold of

• rder m = |R|.

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

A New Piece of Terminology

Why do I call them “scaffolds”?

Image from https://www.bie.org/blog/gold standard pbl scaffold student learning

Others have referred to these as “star-triangle diagrams”. Terwilliger credits Arnold Neumaier for their introduction.

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

Definition, a bit more general . . .

◮ A vector space A ⊆ MatX(C) ◮ A (di)graph G = (V (G), E(G)) ◮ A subset R ⊆ V (G) of “red” nodes, and ◮ a map from edges of G to matrices in A:

w : E(G) → A (edge weights)

◮ a subset F ⊆ V (G) of “fixed” nodes and a fixed function

ψ : F → X The scaffold S(G; R, w; F, ψ) is defined as the quantity S(G; R, w; F, ψ) =

• ϕ:V (G)→X

(∀a∈F)(ϕ(a)=ψ(a))

• e∈E(G)

e=(a,b)

w(e)ϕ(a),ϕ(b)

• r∈R
• ϕ(r).

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

Basic Operations

Deletion: = J Contraction: = I

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

Scaffolds Count Homomorphisms

Example: If A is the adjacency matrix of a simple graph Γ, A = A, and we take R = ∅ and w(e) = A for all e ∈ E(G), then S(G; ∅, w) = |Hom(G, Γ)| is the number of graph homomorphisms from G into Γ. For example, S(K3; ∅, A) = A A A counts labelled triangles in Γ. For |V (G)| = n and w(e) = A(Γ) for e ∈ E(G) and w(e) = J − I − A(Γ) otherwise, tind(G, Γ) = S(Kn, ∅, w) |X|n counts the number of copies of G as induced subgraphs of Γ. (Cf. Lov´ asz)

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

Scaffolds on graphs

For fixed Γ, we can vary “diagram” G and its edge weights w(e).

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

Scaffolds on graphs

For fixed Γ, we can vary “diagram” G and its edge weights w(e). In fact, taking the vector space of all scaffolds, the edge weights in span{I, A} yield the same space of scaffolds as edge weights in A.

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

Scaffolds on graphs

For fixed Γ, we can vary “diagram” G and its edge weights w(e). In fact, taking the vector space of all scaffolds, the edge weights in span{I, A} yield the same space of scaffolds as edge weights in A. For second order scaffolds, where S(G; R, w) itself can be viewed as a matrix, we may then take edge weights from the space of all scaffolds, but this goes no further: we obtain the same vector space.

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

Scaffolds on graphs

For fixed Γ, we can vary “diagram” G and its edge weights w(e). In fact, taking the vector space of all scaffolds, the edge weights in span{I, A} yield the same space of scaffolds as edge weights in A. For second order scaffolds, where S(G; R, w) itself can be viewed as a matrix, we may then take edge weights from the space of all scaffolds, but this goes no further: we obtain the same vector space. The dimension of the vector space of all rth order scaffolds over A where A is the adjacency matrix of graph Γ is equal to the number of orbits of Aut(Γ) on r-tuples of vertices. And we have

• ne basis vector
• (x1,...,xr)∈O

ˆ x1 ⊗ ˆ x2 ⊗ · · · ⊗ ˆ xr for each orbit O.

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

Algebra, Combinatorics and Knot Theory

Donald Higman, Tatsuro Ito, and Fran¸ cois Jaeger

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

Knots

Figure: The unknot (or trivial knot); a left-handed trefoil knot; the figure-eight knot.

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

Spin Models

We seek a pair W +, W − of matrices in MatX(C) satisfying several identities: W +W − = |X|I and = |X| W + W − I = W + W − = D W + W + W − W + W + W −

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

Definition of a Spin Model

= W + I αI = = Dα−1 W + W + = W − I α−1I = = Dα W − W −

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

Link Invariant from a Spin Model

Let G = G(L) be the graph with signed edges defined on the gray regions of a link diagram. Consider the scaffold S(G; ∅, w) =

• ϕ:V (G)→X
• e=(a,b)

w(e)ϕ(a),ϕ(b) where w(e) = W when sgn(e) = + and w(e) = W (−) when sgn(e) = −.

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

Link Invariant from a Spin Model

If we write E(G) = E + ∪ E − where E + contains those edges with a positive sign and E − contains those edges with a negative sign, then we may write the partition function Z(L) := v− 1

2 |V (G)|awr(L)S(G; ∅, w)

as Z(L) = v− 1

2 |V (G)|awr(L) ·

• ϕ:V (G)→X

 

• e=(a,b)∈E +

Wϕ(a),ϕ(b)    

• e=(a,b)∈E −

W (−)

ϕ(a),ϕ(b)

  . Here v = |X| and a is a certain complex number.

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

The Scaffold of a Link Diagram

− − − −

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

A big multilinear map

Jaeger points out that the set of edge weightings of a diagram G forms an algebra and, as we vary the function w : E(G) → A, the map w → S(G; R, w) is a multilinear map from A ⊗ A ⊗ · · · ⊗ A to V ⊗r where V = CX and r = |R|. Elements of A act in an obvious way

• n edges, but also on vertices.

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

Henceforth: A is a (Commutative) Bose-Mesner Algebra

Scaffolds may be useful for other contexts, but for the rest of this talk, A is a Bose-Mesner algebra. Algebraically, a (commutative) association scheme is a vector space of matrices closed under ordinary multiplication, entrywise multiplication, and conjugate-transpose, and containing the identities, I and J, for both multiplications.

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

Two Bases for Bose-Mesner Algebra

Schur idempotents {A0, A1, . . . , Ad} (adjacency matrices) Ai ◦ Aj = δi,jAi AiAj =

d

• k=0

pk

ijAk

matrix idempotents {E0, E1, . . . , Ed} (projections onto eigenspaces) EiEj = δi,jEi Ei ◦ Ej = 1 |X|

d

• k=0

qk

ijEk

where the structure constants qk

ij are called Krein parameters and

we know qk

ij ≥ 0.

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

Definition, Specialized to Association Schemes

Let (X, R) be an association scheme. Suppose we are given

◮ A (di)graph G = (V (G), E(G)) ◮ A subset R ⊆ V (G) of “red” nodes, and ◮ a map from edges of G to matrices in our Bose-Mesner

algebra: w : E(G) → A (edge weights) The scaffold S(G; R, w) is defined as the quantity S(G; R, w) =

• ϕ:V (G)→X
• e∈E(G)

e=(a,b)

w(e)ϕ(a),ϕ(b)

• r∈R
• ϕ(r).

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

Triply Regular Association Schemes

An association scheme is said to be triply regular of ∃T r,s,t

i,j,k

= T r,s,t

i,j,k ·

Aj Ai Ar Ak At As Aj Ai Ak

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

Fundamental Identities for Association Schemes

1) pk

ij = 0 if and only if

0 = Ai Aj Ak 2) qk

ij = 0 if and only if

0 = Ei Ek Ej

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

Locally Strongly Regular Graphs

In 1978, Cameron, Goethals and Seidel proved q1

11 = 0

⇒ ∃λ′ A1 A1 A1 A1 A1 A1 = λ′· A1 A1 A1

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

The Drumstick

Since Ei ◦ Ej =

1 |X|

d

h=0 qh ijEh we have

(Ei ◦ Ej) Ek = qk

ij

|X|Ek. So = = qk

ij

|X|· Ei Ej Ek Ei ◦ Ej Ek Ek And this is zero whenever qk

ij = 0. Our notation here implies that

no edges of G are incident to the center node in the left diagram besides the three edges shown.

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition Spin Models Association Schemes The Drumstick and the Wing

The Wing

Dual to this is the following identity for intersection numbers: since (AjAk) ◦ Ai = d

• e=0

pe

jkAe

• Ai = pi

jkAi,

we have = pi

jk ·

Ai Aj Ak Ai

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Isthmuses Dickie’s Theorem: a∗

j

= 0 implies a∗

1 = 0

Isthmus Lemma (Suzuki)

Lemma

Let (X, R) be a commutative association scheme. (I) If qe

jk · qe ℓm = 0 for all e = h, then

= qh

ℓm

|X| Ej Ek Eℓ Em Ej Ek Eh

(II) If qe

jk · qe ℓm = 0 for all e = h, then = Ej Ek Eℓ Em Ej Ek Eh Eℓ Em

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Isthmuses Dickie’s Theorem: a∗

j

= 0 implies a∗

1 = 0

Suzuki Isthmus Lemma, Proof of (II):

= Ej Ek Eℓ Em Ej Ek I Eℓ Em = d

e=0

Ej Ek Ee Eℓ Em = Ej Ek Eh Eℓ Em

• William J. Martin

Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Isthmuses Dickie’s Theorem: a∗

j

= 0 implies a∗

1 = 0

Dual Isthmus Lemma

Lemma

Assume (X, R) is a commutative association scheme. (I) If pe

hi · pe jk = 0 for all e = ℓ, then

= pℓ

jk ·

Ai Ah Ak Aj Ai Ah Aℓ

(II) If pe

hi · pe jk = 0 for all e = ℓ, then

= Ai Ah Ak Aj Ai Ah Aℓ Ak Aj

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Isthmuses Dickie’s Theorem: a∗

j

= 0 implies a∗

1 = 0

Dickie’s Theorem (1995, generalized by Suzuki 1998)

Theorem (Dickie, Thm. 4.1.1)

Suppose (X, R) is a cometric association scheme with Q-polynomial ordering E0, E1, . . . , Ed. Write a∗

j = qj

• 1j. If 0 < j < d and a∗

j = 0, then a∗ 1 = 0.

Proof:

0 = Ej E1 Ej 0 = |X| b∗

j

· since (E1 ◦ Ej+1)Ej = qj

1,j+1

|X| Ej Ej Ej+1 E1 E1 Ej 0 = |X| b∗

j

· since qe

j,1 · qe 1,j+1 = 0

for any e = j + 1 Ej+1 Ej+1 E1 Ej E1 Ej

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Isthmuses Dickie’s Theorem: a∗

j

= 0 implies a∗

1 = 0

Proof, continued

So 0 = since this is just a linear combination of zero terms. E1 E1 Ej−1 Ej+1 Ej+1 E1

Ej E1

Ej

This is the inner product of our tensor (which is zero) with

, another tensor, order 3.

E1 E1 Ej−1

,

Next, 0 = since qe

1,j−1 · qe 1,j+1 = 0

for any e = j E1 E1 Ej−1 Ej+1

Ej+1

E1

Ej E1

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Isthmuses Dickie’s Theorem: a∗

j

= 0 implies a∗

1 = 0

Proof, continued

Likewise, 0 = since qe

1,j−1 · qe 1,j+1 = 0

for any e = j

E1 E1 Ej−1 Ej+1 Ej+1 E1 E1

Using the entrywise product,

0 = E1 E1 E1 ◦ Ej−1 Ej+1 Ej+1

E1

. Now we expand E1 ◦ Ej−1 =

b∗

j−2

|X| Ej−2 + a∗

j−1

|X| Ej−1 + c∗

j

|X|Ej and observe

qe

1,j+1 = 0 for e < j. Since c∗ j = 0, we have

0 = E1 E1 Ej Ej+1 Ej+1

E1

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Isthmuses Dickie’s Theorem: a∗

j

= 0 implies a∗

1 = 0

End of Sample Proof

0 = since qe

1,j · qe 1,j+1 = 0

for any e = j + 1 E1 E1 Ej Ej+1

E1

Now we have a drumstick!

E1 Ej Ej+1

and we know (Ej ◦ Ej+1)E1 = q1

j,j+1

|X| E1 with q1

j,j+1 = 0 by the cometric property. So we have

0 = E1 E1 E1

So SUM(E1 ◦ E1 ◦ E1) = 0 which tells us that q1

11 = 0, or a∗ 1 = 0.

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Isthmuses Dickie’s Theorem: a∗

j

= 0 implies a∗

1 = 0

The Published Proof

qj

1j = 0

⇒ =

• x∈X

(Ej)a,x(E1)x,b(Ej)x,c (∀a, b, c) =

• x∈X
• y∈X

(Ej)a,y(E1)y,x(Ej+1)y,x(E1)x,b(Ej)x,c (Drumstick) =

• x,y,z∈X

(Ej)a,y(E1)y,z(Ej+1)y,x(Ej+1)x,z(E1)x,b(Ej)z,c so =

• a,b,c,x,y,z∈X

[(E1)a,b(E1)b,c(Ej−1)a,c] · (Ej)a,y(E1)y,z(Ej+1)y,x(Ej+1)x,z(E1)x,b(Ej)z,c

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Isthmuses Dickie’s Theorem: a∗

j

= 0 implies a∗

1 = 0

The Published Proof

=

• a,b,x,y,z∈X

[(E1)a,b(E1)b,z(Ej−1)a,z] · (Ej)a,y(E1)y,z(Ej+1)y,x(Ej+1)x,z(E1)x,b =

• b,x,y,z∈X

[(E1)y,b(E1)b,z(Ej−1)y,z] (E1)y,z(Ej+1)y,x(Ej+1)x,z(E1)x,b =

• b,x,y,z∈X

(E1 ◦ Ej−1)y,z(E1)y,b(E1)b,z(Ej+1)y,x(Ej+1)x,z(E1)x,b =

• b,x,y,z∈X

(Ej)y,z(E1)y,b(E1)b,z(Ej+1)y,x(Ej+1)x,z(E1)x,b =

• b,x,y∈X

(Ej)y,x(E1)y,b(E1)b,x(Ej+1)y,x(E1)x,b =

• b,x∈X

(E1)x,b(E1)b,x(E1)x,b ⇒ 0 = q1

1,1 .

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Isthmuses Dickie’s Theorem: a∗

j

= 0 implies a∗

1 = 0

Sample Result

Let L∗

r =

• qi

rj

• i,j.

Lemma: If, for all e = u, either (L∗

r L∗ s)t,e = 0 or qe ij = 0, then

Ei Et Es Er Ej = Ei Et Es Er Ej Eu

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Isthmuses Dickie’s Theorem: a∗

j

= 0 implies a∗

1 = 0

More Dual Pairs of Identities

A4 A1 A1 A1 A1 A4 A1 A1 A1 A1 A3 A2 A4 A1 A1 A1 A1 A2 A2 E1 E4 E1 E1 E1 E1 E4 E1 E1 E1 E3 E2 E1 E4 E1 E1 E1 E2 E2 William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Isthmuses Dickie’s Theorem: a∗

j

= 0 implies a∗

1 = 0

The Dual S(G †; R†, w †) of a Planar Scaffold S(G; R, w)

Assume G can be drawn in a disk with exactly the red nodes on the boundary. Assume all edge weights are in {A0, . . . , Ad}. Place a vertex of G † in each face. New “red” vertices are those incident to the boundary of the disk. Rotate each edge 90◦ clockwise and replace edge weight Ai by Ei.

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Isthmuses Dickie’s Theorem: a∗

j

= 0 implies a∗

1 = 0

Conjecture: Duality of Planar Scaffolds

Let {Si,j | 1 ≤ i ≤ n, 1 ≤ j ≤ mi} and T1, . . . , Tp be planar scaffolds with edge weights of form Ai and corresponding dual scaffolds S†

i,j and T† 1, . . . , T† p.

For all association schemes with sufficiently many classes, if p

h=1 Th = 0 whenever mi

• h=1

Si,h = 0 ∀1 ≤ i ≤ n then

mi

• h=1

S†

i,h = 0

∀1 ≤ i ≤ n implies p

h=1 T† h = 0.

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Isthmuses Dickie’s Theorem: a∗

j

= 0 implies a∗

1 = 0

The End

Thank You. Monday night’s dinner in Shiogama! (Sushi Tetsu)

William J. Martin Scaffolds