Commutative association schemes

William J. Martin, Hajime Tanaka

Research output: Contribution to journalArticlepeer-review

40 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)1497-1525
Number of pages29
JournalEuropean Journal of Combinatorics
Volume30
Issue number6
DOIs
Publication statusPublished - 2009 Aug

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics

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