Association schemes. Designed experiments, algebra and combinatorics.

*(English)*Zbl 1051.05001
Cambridge Studies in Advanced Mathematics 84. Cambridge: Cambridge University Press (ISBN 0-521-82446-X/hbk). xviii, 387 p. (2004).

The development of combinatorial design theory can be traced back through two main threads. The first leads to finite geometry and algebra, the second to design of statistical experiments. In a series of seminal papers, R.C. Bose and his colleagues unified these two threads, in the process extending the mathematical foundation for design of experiments. In 1952, Bose and Shimamoto formalized one of the central objects in elucidating this foundation, the association scheme. In 1959, Bose and Mesner explored the alegbra underpinning partially balanced designs. This has led to substantial research in many directions in geometry, algebra, combinatorics, experimental design, and coding theory. Unfortunately, the breadth of the research spawned has (perhaps inevitably) resulted in a fragmentation of the research. As the author remarks, much of the current mathematical research is “couched in the language of abstract algebra, with the unfortunate effect that it is virtually inaccessible to statisticians.” By the same token, association schemes are of interest in their own right in algebraic combinatorics, with the (again unfortunate) result that combinatorialists are often unaware of the consequences in experimental design. This book sets out on a very ambitious course, to bring these two groups together. It succeeds admirably. The book starts with a gentle introduction, developing an understanding of the mathematical concept of association schemes and concurrently explaining why these relate to designed experiments. As the exposition develops, the mathematical depth increases – but the rigour of the presentation is consistent throughout. The book is best read from the beginning; it is not designed to “jump in” at an arbitrary place. The mathematical background needed within the first six chapters (the core introduction) is modest, and available to most students at an undergraduate level. More specialized material arises later, but is introduced as needed. In the same way, while a basic understanding of designed experiments makes the core introduction easier, this section can be well understood with minimal prior knowledge in the area. Viewing the book chapter by chapter, the core introduction (Chapters 1–6) develops the main theory. Chapter 1 describes three equivalent ways to view association schemes, and provides instructive examples. Chapter 2 develops the basics of the Bose-Mesner algebra, and the fundamental concept of character tables. In Chapter 3, the notions of crossing and nesting are developed, and shown to provide many further examples. Chapter 4 gives the general theory of incomplete-block designs, which Chapter 5 specializes to partially balanced incomplete-block designs. Then Chapter 6 explores a different view, as partitions of a set, to introduce orthogonal block structures. Chapter 7 explores orthogonal block structures with more than one system of blocks. Chapter 8 delves into generalizations of the material in the first seven chapters, employing more substantial group theory. Chapter 9 develops poset block structures, extending the earlier presentation further into design of experiments. Chapters 10 and 11 form the most mathematical part of the book, focussing on the algebra and statistics. Chapter 12 is a brief look to the future, and Chapter 13 is a brief (but informative) summary of the past. A useful glossary, set of references, and index round out the book. One might feel that a reader interested only in designed experiments, or conversely only in algebraic combinatorics, may find the union of the two more difficult. However the book is written so that both types of reader will find the presentation engaging; indeed one cannot help but conclude that these two viewpoints belong together. The author has done the community a valuable service by strengthening a link between the two.

Reviewer: Charles J. Colbourn (Tempe)

##### MSC:

05-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to combinatorics |

05E30 | Association schemes, strongly regular graphs |

05B05 | Combinatorial aspects of block designs |

05B15 | Orthogonal arrays, Latin squares, Room squares |

62K10 | Statistical block designs |