An algebraic approach to association schemes
This book is devoted to an algebraic approach to association schemes. Let \(X\) be a set, \(1_X=\{(x,x)\mid x\in X\}\), \(r\subseteq X\times X\). We set \(r^*=\{(y,z)\mid(z,y)\in r\}\) and, for each \(x\in X\), we define \(xr=\{y\in X\mid (x,y)\in r\}\). The pair \((X,G)\) is called an association scheme (scheme), if (1) \(G\) is a partition of \(X\times X\) such that \(1_X\in G\) and if \(g\in G\), then \(g^*\in G\); (2) for all \(d,e,f\in G\) there exists a cardinal number \(a_{def}\), such that for all \(y,z\in X\) we have \((y,z)\in f\Rightarrow|yd\cap ze^*|=a_{def}\). This definition of scheme is more general than the usual one (offen requiring that \(|X|\) is finite and \(a_{def}= a_{edf}\)). The elements of \(\{a_{def}\mid d,e,f\in G\}\) will be called the structure constants of \((X,G)\) (this numbers are also called intersection numbers). Similar to the theory of groups, the theory of schemes is viewed as an elementary algebraic theory which is naturally connected to certain geometric structures. In fact, we may view the class of groups and the class of buildings as distinguished classes of schemes. The Coxeter schemes (thin schemes) form the class of those schemes which represent the buildings (the groups) within the class of all schemes. The starting point of the author's approach to schemes is the definition of the complex product in \(G\): for all \(E,F\subseteq G\), \[ EF= \Biggl\{g\in G: \sum_{e\in E} \sum_{f\in F}a_{efg}\neq 0\Biggr\}. \] The first chapter is devoted to elementary consequences of this definition. The second chapter considers the theory of ``composition factors''. After that ``direct'', ``quasi-direct'' and ``semi-direct'' products are introduced. In the third chapter, the author collects various algebraic results which are needed for the representation theory (chapter 4) and for the theory of generators of schemes (chapter 5). The fourth chapter gives a general introduction into the representation theory of finite schemes, in particular a generalization of the H. Maschke theorem on the semisimplicity of group algebras. The fifth chapter may be consider as the heart of this book. In this chapter constrained schemes are investigated. This class of schemes is slightly larger then the above-mentioned class of Coxeter schemes (in particular, buildings and thick constrained schemes are the same thing). In this chapter the author gives a complete proof of the W. Feit and G. Higman theorem on finite generalized polygons and a simultaneous approach to the theorems of S. Paine and U. Ott on polarities of finite generalized quadrangles and of finite generalized hexagons. The main focus of the remainder of the fifth chapter is then on an appropriate generalization of Coxeter schemes of rank 2. No attempt has been made to give a complete account of the results available on schemes. The well-founded connection between schemes and graphs has been omitted completely.
scientific article; zbMATH DE number 926846