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 orderings ◮ 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 m 1 = Q 01 > Q 11 > · · · > Q d 1 , A = � A 1 � ◮ √ m 1 ≤ v 1 ≤ τ m − 1 and m 1 ≤ v 1 + v i whenever i > 1 with p i 11 > 0 ◮ “Semidefinite programming”: Kodalen/WJM/Yu apply Schoenberg’s Theorem. Evaluating the Gegenbauer polynomial G d +2 ( t ) on E 1 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 | X | d struc multiplicities valencies G , H 3 1288 P 1 , 22 , 230 , 1035 M 23 , M 11 1 , 165 , 330 , 792 4 11178 P Co 3 , HS 1 , 23 , 275 , 2024 , 8855 1 , 1100 , 5600 , 4125 , 352 2Q (02431) 4 13056 P 1 , 135 , 3400 , 8925 , 595 Sp(8 , 2) , S 10 1 , 210 , 1575 , 5600 , 5670 O + 8 (3) . 2 , O + 4 28431 P 1 , 260 , 9450 , 18200 , 520 8 (2) . 2 1 , 960 , 3150 , 22400 , 1920 5 352 A M 22 . 2 , A 7 1 , 21 , 154 , 154 , 21 , 1 1 , 35 , 105 , 126 , 70 , 15 5 28160 A Fi 22 .2 , O 7 (3) 1 , 429 , 13650 , 13650 , 429 , 1 1 , 364 , 3159 , 12636 , 10920 , 1080 PSO − (10 , 2) , S 12 5 104448 P 1 , 187 , 7700 , 56100 , 39270 , 1190 1 , 462 , 5775 , 30800 , 62370 , 5040 6 704 AB 1 , 22 , 175 , 308 , 175 , 22 , 1 HS.2 , U 3 (5) 1 , 50 , 175 , 252 , 175 , 50 , 1 2Q (0523416) 7 4050 A McL.2 , M 22 1 , 22 , 252 , 1750 , 1750 , 252 , 22 , 1 1 , 176 , 462 , 1155 , 1232 , 672 , 330 , 22 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 3 3 − 2 0 1 / 3 0   − 2 M M   = − 2 = − 6 ◮ − 2   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 , � � ⊤ N = 2 / 3 2 / 3 1 / 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 − 2 3 3 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

Definition Examples Spin Models 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 vector subspace of Mat X ( C ). Our standard basis for V = C X 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 ) = 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

Definition Examples Spin Models 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”? 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

Definition Examples Spin Models Definitions and Basics Association Schemes Some Results of Dickie and Suzuki The Drumstick and the Wing Definition, a bit more general . . . ◮ A vector space 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 ) . r ∈ R ϕ : V ( G ) → X e ∈ E ( G ) ( ∀ a ∈ F )( ϕ ( a )= ψ ( a )) e =( a , b ) William J. Martin Scaffolds

Definition Examples Spin Models 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

Definition Examples Spin Models 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 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( K 3 ; ∅ , 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, t ind ( G , Γ) = S( K n , ∅ , w ) | X | n counts the number of copies of G as induced subgraphs of Γ. (Cf. Lov´ asz) William J. Martin Scaffolds

Definition Examples Spin Models Definitions and Basics Association Schemes Some Results of Dickie and Suzuki 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

Definition Examples Spin Models Definitions and Basics Association Schemes Some Results of Dickie and Suzuki 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

Definition Examples Spin Models Definitions and Basics Association Schemes Some Results of Dickie and Suzuki 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