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Scaffolds: A graph-theoretic tool for tensor computations related to Bose-Mesner algebras.
- Source :
-
Linear Algebra & its Applications . Jun2021, Vol. 619, p50-106. 57p. - Publication Year :
- 2021
-
Abstract
- We introduce a pictorial notation for certain tensors arising in the study of association schemes, based on earlier ideas of Terwilliger, Neumaier and Jaeger. These tensors, which we call "scaffolds", obey a simple set of rules which generalize common linear-algebraic operations such as trace, matrix product and entrywise product. We first study an elementary set of "moves" on scaffolds and illustrate their use in combinatorics. Next we re-visit results of Dickie, Suzuki and Terwilliger. The main new results deal with the relationships among vector spaces of scaffolds with edge weights chosen from a fixed coherent algebra and various underlying diagrams. As one consequence, we provide simple descriptions of the Terwilliger algebras of triply regular and dually triply regular association schemes. We finish with a conjecture connecting the duality of Bose-Mesner algebras to the graph-theoretic duality of circular planar graphs. [ABSTRACT FROM AUTHOR]
- Subjects :
- *ALGEBRA
*VECTOR spaces
*MATRIX multiplications
*PLANAR graphs
*COMBINATORICS
Subjects
Details
- Language :
- English
- ISSN :
- 00243795
- Volume :
- 619
- Database :
- Academic Search Index
- Journal :
- Linear Algebra & its Applications
- Publication Type :
- Academic Journal
- Accession number :
- 149549872
- Full Text :
- https://doi.org/10.1016/j.laa.2021.02.009