Preface to the Japanese version

The purpose of this book is to give an introduction to algebraic combinatorics. There is no explicit definition of algebraic combinatorics. In Algebraic Combinatorics I (1984), written by Eiichi Bannai and Tatsuro Ito, algebraic combinatorics is described as “a group theorywithout groups” or “a character theoretical study of combinatorial objects.” Specifically speaking, we pursue the study of combinatorics as an extension or a generalization of the study of finite permutation groups. In this book, we keep this direction in our mind. This is also the approach, initiated by Philippe Delsarte, which enables us to look at a wide range of combinatorial objects such as graphs, codes, and designs from a unified viewpoint. Based on these thoughts, we explain Delsarte’s theory and its various extensions. When we finished writing Algebraic Combinatorics I, we wanted to write the sequel Algebraic Combinatorics II soon. We regret we did not write it up at that time, but due to various reasons, we could not. We would not say that it has nothing to do with our laziness, but if we could use an excuse, it seems that the developments in the field are not enough to complete a book, and the range of mathematical objects that we are interested in has expanded too widely to handle. Therefore, we thought we could and should write on algebraic combinatorics on spheres, and in 1999, Eiichi Bannai and Etsuko Bannai published Algebraic Combinatorics on Spheres, written in Japanese. We did not prepare an English edition because we planned to write Algebraic Combinatorics II in the future, and we regard the above book as a preparation for it. In the present book, we want to present another subject, Delsarte’s theory in association schemes. Besides, we would like to start writing on the classification of Pand Q-polynomial association schemes, which is themost important problem in Algebraic Combinatorics I, through the study of Terwilliger algebras by Terwilliger and Ito. In this sense, writing up this book enabled us to work seriously on Algebraic Combinatorics II. Before we know it, we get older. Counting the remaining time in our lives, we would like to write one more work. We give an overview on the contents of this book in the following. Chapter 1 is the introduction to classical combinatorics. The contents are suitable for undergraduate courses. In Japan, there are not so many universities offering lectures on combinatorics for undergraduates. Therefore, we selected basic subjects which beginners in combinatorics should learn. The contents are slightly long for a one-semester course. Chapter 2 is the introduction to association schemes. Some contents overlap with Algebraic Combinatorics I (and with Algebraic Combinatorics on Spheres). We start with the basics, so the chapter is comprehensive for readers who are not familiar with this area. Chapter 3 is the introduction to Delsarte’s theory, which is the theory of codes and designs in association schemes. The description is faithful to the original paper by Delsarte (1973). The contents up to this chapter are understandable to undergraduate students. Chapter 4 is the extension of Delsarte’s theory. Our aim is to introduce