An Algebraic Approach to Association Schemes
The primary object of the lecture notes is to develop a treatment of association schemes analogous to that which has been so successful in the theory of finite groups. The main chapters are decomposition theory, representation theory, and the theory of ge
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Paul-Hermann Zieschang
An Algebraic Approach to Association Schemes
Springer
Author Paul-Hermann Zieschang Christian-Albrechts-Universitat zu Kiel Mathematisches Seminar Ludewig-Meyn-StraBe 4 24098 Kiel, Germany
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Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Zieschang, Paul-Hermann: An algebraic approach to association schemes / Paul-Hermann Zieschang. - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London ; Milan; Paris ; Tokyo : Springer, 1996 (Lecture notes in mathematics ; 1628) ISBN 3-540-61400-1 NE:GT Mathematics Subject Classification (1991): 05-02, 05E30, 51E 12, 5lE24, 05B25, 16S50 ISSN 0075-8434 ISBN 3-540-61400-1 Springer-Verlag Berlin Heidelberg New York
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Introduction
Let X be a set. We define
Ix := {(x,x) Let r
I x EX}.
X x X be given. We set
r*:= {(y,z)
I (z,y)
E r},
and, for each x E X, we define
xr:={yEX I (x,Y)Er}. Let G be a partition of X x X such that 0 rt G :1 Ix, and assume that, for each 9 E G, s' E G. Then the pair (X, G) will be called an association scheme if, for all d, e, f E G, there exists a cardinal number adej such that, for all u, z EX, (y,z) E f => Iydn ze*1 = adej.l In these notes, we shall always say scheme instead of association scheme. The pair (X, G) will always denote a scheme. We shall always write I instead of Ix. The elements of {adej I d, e, f E G} will be called the structure constants of (X, G).2 Occasionally we shall use the expression regularity condition in order to denote the condition which guarantees the existence of the structure constants. The present text provides an algebraic approach to schemes. Similar to the theory of groups, the theory of schemes will be viewed as an elementary 1
2
This definition of association schemes is slightly more general than the usual one. Usually one requires additionally at least that IXI be finite. In this case, the term "homogeneous coherent configuration" is common, too; see [14]. (Also in the present text the finiteness of IXI plays an important role.) It is also often required that, for all d, e, f E G, ade! = aed!; see, e.g., [1]. The term "association scheme" was introduced in
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