Scaffolds: a graph-based system for computations in Bose-Mesner algebras

Abstract

Let X be a finite set and let Mat X (C) denote the algebra of matrices with rows and columns indexed by X and entries from the complex numbers acting on C X with standard basis {x | x ∈ X}. For a digraph G = (V (G), E(G)), function R : [m] → V (G) with r j := R(j), and a function w from the arcs of G to Mat X (C), we define the "scaffold" S(G, R; w) as the sum over all functions ϕ from V (G) to X of the m-fold tensors ϕ(r 1 )⊗ ϕ(r 2 )⊗• • •⊗ ϕ(r m ) scaled by the product of the entries w(e) ϕ(a),ϕ(b) over all arcs e = (a, b) of G. If Γ is a digraph with adjacency matrix A, m = 0 and w(e) = A for all e, this simply counts all digraph homomorphisms from G to Γ. If instead, Γ is a distance-regular graph with i th distance matrix A i and G is a star graph K 1,m with central node a and degree one nodes r 1 , . . . , r m , the scaffold S(G, R; w) encodes generalized intersection numbersScaffolds also arise in the the theory of link invariants and spin models: the partition function of a link diagram is encoded as a scaffold of order zero and invariance under the three Reidemeister moves are encoded as identities among scaffolds of order two and three. These diagrams were introduced in the late 1980s by Arnold Neumaier to aid in computations in the theory of distance-regular graphs. Various authors have used scaffolds to systematize complex calculations of parameters in association schemes. Prioritizing the tutorial value of the paper, we briefly revisit results of Dickie, Suzuki, and Terwilliger using this diagrammatic formalism. Commonly employed transformations are presented as a basic system of "moves" on these diagrams that preserve their value and certain natural actions of the Bose-Mesner algebra on nodes and edges of a scaffold given by Terwilliger and Jaeger are reviewed. Many of the results presented here are not new; our goal is to collect and present, in a uniform fashion, Neumaier's original idea extended to tensors and its use by various authors. Sometimes the term "star-triangle diagram" appears for what we, in this paper, call "scaffolds".When one fixes the diagram G and root nodes R but allows the edge weights to vary over matrices in a given coherent algebra A, the vector space W((G, R); A) spanned by all resulting tensors seems worthy of study. We show that W((H, R ′ ); A) is contained in W((G, R); A) when (H, R ′ ) is a rooted minor of (G, R) and examine several important spaces of this form with three root nodes in connection with the Terwilliger algebra.It is not surprising that some scaffold identities are more intuitive than others: for instance, much more is known about distance-regular graphs than is known about Q-polynomial ("cometric") association schemes. Connecting duality in the theory of association schemes to duality of circular planar graphs, we present a conjecture dealing with dual pairs of planar scaffolds which points to a tool for the generation of new identities.

A duality of scaffolds for translation association schemes

“…Ei where E i , E j , and E k are primitive idempotents of an association scheme. Martin [7] called these tensors (and also their diagrams) scaffolds, and studied them systematically and comprehensively. The red nodes are referred to as the root nodes or simply roots.…”

“…Moreover, the validity of these dual implications was usually verified by also dualizing the proofs. Martin [7,Conjecture 4.1] conjectured that this process is always possible. Namely, his conjecture asserts that if an implication among scaffold equations with edge weights from the A i is true for all association schemes, then so is the dual implication.…”

“…Later we will assume that the set X is the vertex set of an association scheme, so we call the elements of V nodes to distinguish V from X. The (symmetric) scaffold S(G, R; w) of order ℓ [7] is the vector in C X ℓ = (C X ) ⊗ℓ defined by…”

“…w(e) t(e) σ ,h(e) σ r σ , where we write r σ := r σ 1 ⊗ • • • ⊗ r σ ℓ for brevity. Martin [7] also considered the case where σ ranges over a certain subset of X V , but we will only discuss the above unrestricted case in this paper.…”

“…We also allow that the scaffolds in different sequences have different orders. Second, we dropped the condition in [7,Conjecture 4.1] about bijections among the sets of root nodes of the scaffolds. This condition seems unnecessary because the roots are already ordered by definition.…”