Commutative association schemes

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.Comment: 36 page

Permutation group approach to association schemes

AbstractWe survey the modern theory of schemes (coherent configurations). The main attention is paid to the schurity problem and the separability problem. Several applications of schemes to constructing polynomial-time algorithms, in particular, graph isomorphism tests, are discussed

有限射影幾何の組合せ構造の代数的性質について

Tohoku University田中太初課

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

Combinatorial Extensions of Terwilliger Algebras and Wreath Products of Association Schemes

We introduce the notion of the combinatorial extension of a Terwilliger algebra by a coherent algebra. By using this notion, we find a simple way to describe the Terwilliger algebras of certain coherent configurations as combinatorial extensions of simpler Terwilliger algebras. In particular, given an association scheme SS and another association scheme RR such that the Terwilliger algebra of RR is isomorphic to a coherent algebra, we prove that the Terwilliger algebra of the wreath product S≀R is isomorphic to the combinatorial extension of the Terwilliger algebra of SS by a coherent algebra. We also show that the Terwilliger algebra of the wreath product WW of rank 22 association schemes can be expressed as the combinatorial extension of adjacency algebras of association schemes induced by the closed subsets of WW. As a direct consequence, we obtain simple conceptual explanations and alternative proofs of many known results on the structures of Terwilliger algebras of wreath products of association schemes