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AGT: Scaffolds

Talk by Bill Martin. Building on the work of various authors who have used tensors in the study of association schemes and spin models, I propose the term “scaffold” for certain tensors that have been represented by what are sometimes called “star-triangle diagrams” in the literature. The main goal of the talk is to introduce and motivate these objects which somewhat resemble partition functions as they appear in combinatorics. (The exact definition is too cumbersome to include here.) Certain manipulations of these tensors follow an intuitive set of rules which we outline mostly without proof. We touch on connections to association schemes, spin models and homomorphism densities before focusing on coherent algebras. A coherent algebra on a non-empty set X is a vector subspace of the algebra of matrices with rows and columns indexed by X and entries from the complex numbers which is closed under the conjugate transpose operation, closed under both ordinary and entrywise multiplication of matrices, and contains the identities for both. Commutative coherent algebras are known as Bose-Mesner alegbras. Given a coherent algebra, we explore the vector space spanned by all scaffolds with a given diagram and edge weights chosen from this algebra. A number of elementary concepts from graph theory play a role in the study of these vector spaces, including minors, series-parallel reduction, Delta-Wye transformations, and circular planar duality.