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Algebraic decompositions of commutative association schemes. (English) Zbl 0573.20051
A commutative association scheme of class d is a pair (X,R) of a finite set of points X and R $$=\{R_ 0,R_ 1,...,R_ d\}$$ of non-empty subsets $$R_ i$$ of $$X\times X$$ with the following properties: (i) $$R_ 0=\{(x,x)$$; $$x\in X\}$$, (ii) For every (x,y)$$\in X\times X$$, $$(x,y)\in R_ i$$ for exactly one i. (iii) For each $$i\in \{0,1,...,d\}$$, $${}^ TR_ i=\{(y,x)$$; $$(x,y)\in R_ i\}=R_ j$$ for some $$j\in \{0,1,...,d\}$$. (iv) For each i,j,k$$\in \{0,1,...,d\}$$, $$| \{z\in X$$; $$(x,z)\in R_ i$$, $$(z,y)\in R_ j\}| =p^ k_{ij}=const.$$, whenever $$(x,y)\in R_ k$$. (v) $$p^ k_{ij}=p^ k_{ji}.$$
The connection between finite groups and commutative association schemes is known. Using this connetion the authors present certain constructions and results for commutative association schemes which arise in group theory. In Section 1, the authors define direct products of commutative association schemes and they relate the incidence matrices and parameters of two commutative association schemes with those of their direct product. In Section 2, the authors investigate the properties of commutative association schemes which are analogous to some properties of abelian groups. Finally, in Section 3, the authors prove the Krull- Schmidt theorem for commutative association schemes.
Reviewer: K.Burian

##### MSC:
 20K01 Finite abelian groups 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.) 20D60 Arithmetic and combinatorial problems involving abstract finite groups 15A30 Algebraic systems of matrices
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