Commutative association schemes

TL;DR

This survey reviews recent advances in commutative association schemes, highlighting developments in Gelfand pairs, cometric schemes, Delsarte Theory, spin models, and semidefinite programming, emphasizing the interplay between combinatorial and group-theoretic symmetries.

Contribution

It synthesizes recent progress in the theory of commutative association schemes, proposing the Terwilliger algebra as a key tool for future research.

Findings

Schrijver's SDP bound for binary codes explained

Connection between Terwilliger algebra and Delsarte Theory highlighted

Recent developments in Gelfand pairs and cometric schemes summarized

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.