An algebraic approach to association schemes
type: Item
instance of: scholarly article
title: An algebraic approach to association schemes
zbMATH Open document ID: 0857.05100
author: Paul-Hermann Zieschang
published in: Lecture Notes in Mathematics
publication date: 1996-09-15T00:00:00Z
review text: 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 particu…
Mathematics Subject Classification ID: 05E30
Mathematics Subject Classification ID: 05-02
Mathematics Subject Classification ID: 51E12
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 926846
zbMATH Keywords: generalized polygons
zbMATH Keywords: association schemes
zbMATH Keywords: structure constants
zbMATH Keywords: intersection numbers
zbMATH Keywords: buildings
zbMATH Keywords: Coxeter schemes
zbMATH Keywords: complex product
zbMATH Keywords: representation theory
zbMATH Keywords: generators of schemes
zbMATH Keywords: Maschke theorem
zbMATH Keywords: semisimplicity
zbMATH Keywords: group algebras
zbMATH Keywords: constrained schemes
zbMATH Keywords: finite generalized polygons
zbMATH Keywords: polarities
zbMATH Keywords: generalized quadrangles
zbMATH Keywords: generalized hexagons
reviewed by: Alexandre A. Makhnev
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0097032
OpenAlex ID: W599879082
DOI: 10.1007/BFB0097032
description: scientific article; zbMATH DE number 926846