Scaffolds A graph-based system for computations in Bose-Mesner - - PowerPoint PPT Presentation

Examples Definitions and Basics Some Results of Dickie and Suzuki

Scaffolds

A graph-based system for computations in Bose-Mesner algebras William J. Martin

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

Algebraic Combinatorics Seminar Shanghai Jiao Tong University, Shanghai October 18, 2016

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

=

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  

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

=

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

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

=

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  

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

=

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

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

• 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

=

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

=

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 Association Schemes The Drumstick and the Wing

Definition

Let X be a finite set and let A be a subalgebra of MatX(C). 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 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 Association Schemes The Drumstick and the Wing

A New Piece of Terminology

Why do I call them “scaffolds”? 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 Association Schemes The Drumstick and the Wing

Definition, a bit more general . . .

◮ 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 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 Association Schemes The Drumstick and the Wing

Scaffolds Count Homomorphisms

Example: If A is the adjacency matrix of a simple graph H, A = A, and we take R = ∅ and w(e) = A for all e ∈ E(G), then S(G; ∅, w) = |Hom(G, H)| is the number of graph homomorphisms from G into H. For example, S(K3; ∅, A) = A A A counts labelled triangles in H.

William J. Martin Scaffolds

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

Partition Functions

◮ statistical mechanics ◮ graph theory ◮ E.g., Tutte polynomial is partition function of the Potts model ◮ Vaughan Jones: spin models (here X is a set of “spins”) yield

link invariants (w(e) = W ±) Triply Regular Association Scheme: ∃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 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 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 Association Schemes The Drumstick and the Wing

Commutative Association Schemes

Combinatorially, an association scheme is an ordered pair (X, R) where X is a finite set and R = {R0, . . . , Rd} is a partition of X into binary relations satisfying

◮ R contains the identity relation: R0 = {(a, a) | a ∈ X} ◮ for each i, there is an i′ ∈ {0, . . . , d} such that

Ri′ = R⊤

i

= {(b, a) | (a, b) ∈ Ri}

◮ there exist constants pk ij such that, whenever (a, b) ∈ Rk,

|{c ∈ X | (a, c) ∈ Ri, (c, b) ∈ Rj}| = pk

ij ◮ pk ij = pk ji

William J. Martin Scaffolds

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

The Association Scheme of Symmetric Group S3

X = { (1), (12), (13), (23), (123), (132) }

(23) (13) (12) (132) (123) (1) (23) (13) (12) (132) (123) (1)

One Cayley graph for each conjugacy class C0 = {(1)}, C1 = {(12), (13), (23)}, C2 = {(123), (132)}

William J. Martin Scaffolds

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

The Association Scheme of S3

A0 =        

1 1 1 1 1 1

        , A1 =        

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

        , A2 =        

1 1 1 1 1 1 1 1 1 1 1 1

       

William J. Martin Scaffolds

Examples Definitions and Basics Some Results of Dickie and Suzuki Definition 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 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 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 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 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

P- and Q-Polynomial Schemes

A (symmetric) association scheme (X, R) with relations {Ri} is P-polynomial (or “metric”) with respect to the ordering A0, . . . , Ad

• f its Schur idempotents if, with respect to this ordering,

k > i + j ⇒ pk

ij = 0

and k = i + j ⇒ pk

ij = 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

P- and Q-Polynomial Schemes

A (symmetric) association scheme (X, R) with relations {Ri} is P-polynomial (or “metric”) with respect to the ordering A0, . . . , Ad

• f its Schur idempotents if, with respect to this ordering,

k > i + j ⇒ pk

ij = 0

and k = i + j ⇒ pk

ij = 0

A (symmetric) association scheme (X, R) with primitive idempotents {Ej} is Q-polynomial (or “cometric”) with respect to the ordering E0, . . . , Ed if, with respect to this ordering, k > i + j ⇒ qk

ij = 0

and k = i + j ⇒ qk

ij = 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

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 End

Thank You. Special thanks to Professor Eiichi Bannai, Professor Yaokun Wu, and to Yan Zhu.

William J. Martin Scaffolds