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 Introduction Definitions and Basics Exercises Some Results of Dickie and Suzuki 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 ◮ N ◮ Matrix N , as a second-order tensor, is represented as N ◮ The sum of all entries of N is ◮ The ordinary matrix product of M and N is encoded as a M N series reduction M ◮ Entrywise multiplication is a parallel reduction N . William J. Martin Scaffolds
Examples Introduction Definitions and Basics Exercises Some Results of Dickie and Suzuki Example − 2 3 3 0 1 / 3 1 / 3 , N = X = { 1 , 2 , 3 } , M = 3 − 2 3 1 / 3 0 1 / 3 − 2 3 3 0 1 / 3 0 M = ◮ William J. Martin Scaffolds
Examples Introduction Definitions and Basics Exercises Some Results of Dickie and Suzuki Example − 2 3 3 0 1 / 3 1 / 3 , N = X = { 1 , 2 , 3 } , M = 3 − 2 3 1 / 3 0 1 / 3 − 2 3 3 0 1 / 3 0 − 2 M = − 2 ◮ − 2 William J. Martin Scaffolds
Examples Introduction Definitions and Basics Exercises Some Results of Dickie and Suzuki Example − 2 3 3 0 1 / 3 1 / 3 , N = X = { 1 , 2 , 3 } , M = 3 − 2 3 1 / 3 0 1 / 3 − 2 3 3 0 1 / 3 0 − 2 M = − 2 ◮ − 2 M = ◮ William J. Martin Scaffolds
Examples Introduction Definitions and Basics Exercises Some Results of Dickie and Suzuki Example − 2 3 3 0 1 / 3 1 / 3 , N = X = { 1 , 2 , 3 } , M = 3 − 2 3 1 / 3 0 1 / 3 − 2 3 3 0 1 / 3 0 − 2 M = − 2 ◮ − 2 M = − 6 ◮ William J. Martin Scaffolds
Examples Introduction Definitions and Basics Exercises Some Results of Dickie and Suzuki Example − 2 3 3 0 1 / 3 1 / 3 , N = X = { 1 , 2 , 3 } , M = 3 − 2 3 1 / 3 0 1 / 3 − 2 3 3 0 1 / 3 0 − 2 M = − 2 ◮ − 2 M = − 6 ◮ N = ◮ William J. Martin Scaffolds
Examples Introduction Definitions and Basics Exercises Some Results of Dickie and Suzuki Example − 2 3 3 0 1 / 3 1 / 3 , N = X = { 1 , 2 , 3 } , M = 3 − 2 3 1 / 3 0 1 / 3 − 2 3 3 0 1 / 3 0 − 2 M = − 2 ◮ − 2 M = − 6 ◮ 0 1 / 3 1 / 3 N = 1 / 3 0 1 / 3 ◮ 0 1 / 3 0 William J. Martin Scaffolds
Examples Introduction Definitions and Basics Exercises Some Results of Dickie and Suzuki Example − 2 3 3 0 1 / 3 1 / 3 , N = X = { 1 , 2 , 3 } , M = 3 − 2 3 1 / 3 0 1 / 3 − 2 3 3 0 1 / 3 0 − 2 M = − 2 ◮ − 2 M = − 6 ◮ 0 1 / 3 1 / 3 N = 1 / 3 0 1 / 3 ◮ 0 1 / 3 0 N ◮ while = William J. Martin Scaffolds
Examples Introduction Definitions and Basics Exercises Some Results of Dickie and Suzuki Example − 2 3 3 0 1 / 3 1 / 3 , N = X = { 1 , 2 , 3 } , M = 3 − 2 3 1 / 3 0 1 / 3 − 2 3 3 0 1 / 3 0 − 2 M = − 2 ◮ − 2 M = − 6 ◮ 0 1 / 3 1 / 3 N = 1 / 3 0 1 / 3 ◮ 0 1 / 3 0 N ◮ while = 5 / 3 William J. Martin Scaffolds
Examples Introduction Definitions and Basics Exercises Some Results of Dickie and Suzuki Example − 2 3 3 0 1 / 3 1 / 3 , N = X = { 1 , 2 , 3 } , M = 3 − 2 3 1 / 3 0 1 / 3 − 2 3 3 0 1 / 3 0 − 2 M = − 2 ◮ − 2 M = − 6 ◮ 0 1 / 3 1 / 3 N = 1 / 3 0 1 / 3 ◮ 0 1 / 3 0 N ◮ while = 5 / 3 N ◮ But � � = 1 / 3 2 / 3 2 / 3 William J. Martin Scaffolds
Examples Introduction Definitions and Basics Exercises Some Results of Dickie and Suzuki Example, continued − 2 3 3 0 1 / 3 1 / 3 , N = X = { 1 , 2 , 3 } , M = 3 − 2 3 1 / 3 0 1 / 3 3 3 − 2 0 1 / 3 0 M N ◮ Products: = William J. Martin Scaffolds
Examples Introduction Definitions and Basics Exercises Some Results of Dickie and Suzuki Example, continued − 2 3 3 0 1 / 3 1 / 3 , N = X = { 1 , 2 , 3 } , M = 3 − 2 3 1 / 3 0 1 / 3 3 3 − 2 0 1 / 3 0 1 1 / 3 1 / 3 M N ◮ Products: = − 2 / 3 2 1 / 3 1 1 / 3 2 M ◮ while N = William J. Martin Scaffolds
Examples Introduction Definitions and Basics Exercises Some Results of Dickie and Suzuki Example, continued − 2 3 3 0 1 / 3 1 / 3 , N = X = { 1 , 2 , 3 } , M = 3 − 2 3 1 / 3 0 1 / 3 3 3 − 2 0 1 / 3 0 1 1 / 3 1 / 3 M N ◮ Products: = − 2 / 3 2 1 / 3 1 1 / 3 2 M 0 1 1 ◮ while N = 1 0 1 0 1 0 William J. Martin Scaffolds
Examples Definition Definitions and Basics Association Schemes Some Results of Dickie and Suzuki The Drumstick and the Wing Definition Let X be a finite set and let A be a subalgebra of Mat X ( 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 ) = w ( e ) ϕ ( a ) ,ϕ ( b ) ϕ ( r ) . r ∈ R ϕ : V ( G ) → X e ∈ E ( G ) e =( a , b ) This is an element of V ⊗| R | , so we say S( G ; R , w ) is a scaffold of order m = | R | . William J. Martin Scaffolds
Examples Definition Definitions and Basics Association Schemes Some Results of Dickie and Suzuki 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 Definition Definitions and Basics Association Schemes Some Results of Dickie and Suzuki The Drumstick and the Wing Definition, a bit more general . . . ◮ A ⊆ Mat X ( 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 ) = w ( e ) ϕ ( a ) ,ϕ ( b ) ϕ ( r ) . ϕ : V ( G ) → X e ∈ E ( G ) r ∈ R ( ∀ a ∈ F )( ϕ ( a )= ψ ( a )) e =( a , b ) William J. Martin Scaffolds
Examples Definition Definitions and Basics Association Schemes Some Results of Dickie and Suzuki The Drumstick and the Wing Basic Operations J = Deletion: I = Contraction: William J. Martin Scaffolds
Examples Definition Definitions and Basics Association Schemes Some Results of Dickie and Suzuki 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( K 3 ; ∅ , A ) = A A A counts labelled triangles in H . William J. Martin Scaffolds
Examples Definition Definitions and Basics Association Schemes Some Results of Dickie and Suzuki 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 ± ) ∃ T r , s , t Triply Regular Association Scheme: i , j , k A s A r A k A i A t = T r , s , t i , j , k · A i A k A j A j William J. Martin Scaffolds
Examples Definition Definitions and Basics Association Schemes Some Results of Dickie and Suzuki The Drumstick and the Wing Algebra, Combinatorics and Knot Theory Donald Higman, Tatsuro Ito, and Fran¸ cois Jaeger William J. Martin Scaffolds
Examples Definition Definitions and Basics Association Schemes Some Results of Dickie and Suzuki 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
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