Commutative association schemes

Abstract

Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.

36 pages

References in corpus (18)

• New upper bounds for kissing numbers from semidefinite programming
• Symmetry in semidefinite programs
• Matrix Algebras and Semidefinite Programming Techniques for Codes
• Some algebra related to $P$-and $Q$-polynomial association schemes
• Real Mutually Unbiased Bases
• Pseudocyclic association schemes and strongly regular graphs
• The Impossibility of Pseudo-Telepathy Without Quantum Entanglement
• New proofs of the Assmus-Mattson theorem based on the Terwilliger algebra
• The Drinfel'd polynomial of a tridiagonal pair
• Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets
• Coding Theory and Algebraic Combinatorics
• A bilinear form relating two Leonard systems
• Colouring an Orthogonality Graph
• Tridiagonal pairs of $q$-Racah type
• Explicit Ramsey graphs and Erdos distance problem over finite Euclidean and non-Euclidean spaces
• Group-case commutative association schemes and their character tables
• Ramanujan Graphs on Cosets of $PG{L}_2({\mathbb{F}}_q)$
• Jones Pairs

Cited by in corpus (26)

• Distance-regular graphs
• On almost distance-regular graphs
• Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems
• Perron Spectratopes and the Real Nonnegative Inverse Eigenvalue Problem
• On relative $t$-designs in polynomial association schemes
• Strongly walk-regular graphs
• Nonexistence of exceptional imprimitive Q-polynomial association schemes with six classes
• Association schemes on general measure spaces and zero-dimensional Abelian groups
• Lattices and Norton algebras of Johnson, Grassmann and Hamming graphs
• Cometric Association Schemes
• Cameron-Liebler $k$-sets in $\text{AG}(n,q)$
• Multimarked Spatial Search by Continuous-Time Quantum Walk
• An Assmus-Mattson theorem for codes over commutative association schemes
• Modular Gelfand pairs and multiplicity-free representations
• Linear programming bounds for regular graphs
• Vertex subsets with minimal width and dual width in $Q$-polynomial distance-regular graphs
• Semidefinite programming bounds for error-correcting codes
• Symmetric bilinear forms over finite fields with applications to coding theory
• A duality of scaffolds for translation association schemes
• The Erdős-Ko-Rado basis for a Leonard system
• Tight relative $t$-designs on two shells in hypercubes, and Hahn and Hermite polynomials
• Scaffolds: a graph-based system for computations in Bose-Mesner algebras
• Association schemes and Hypergroups
• Partially metric association schemes with a multiplicity three
• On $Q$-polynomial association schemes of small class
• Nonexistence of Exceptional 5-class Association Schemes with Two $Q$-polynomial Structures