Imprimitive, Symmetric Association Schemes Of Rank6

In the imprimitive, symmetric association schemes, there is a disconnected graph ofsome association relation. The equivalence relation decided by that gragh is a union ofsome association relations. The equivalence class relating to the equivalence relation iscalled block. According to the theory of association schemes, we can get the parametersof the association subschemes and the quotient association schemes from the parametersof the primitive association schemes.In1995, D.G.Higman has investigated imprimitive, symmetric association schemesof rank5. He has got the parameters of the primitive association schemes from theparameters of the association subschemes or the quotient association schemes and foundexamples. In this paper, we investigate imprimitive, symmetric association schemes ofrank6. Research how to increase variables as little as possible to get the parametersof the primitive association schemes when the parameters of the association subschemes(or the quotient association schemes) are got, such that primitive association schemes arenot wreath products of the association subschemes by the quotient association schemes.In this paper, according to the equivalence relation, the imprimitive, symmetricassociation schemes of rank6are divided into about6classes (Ⅰ，Ⅱ，Ⅲ，Ⅳ，Ⅴ，Ⅵ), We investigate the first two classes. First of all, the intersection matrices and character-multiplicity table of the association subschemes (or the quotient association schemes)are represented by variables as little as possible. Secondly, because of the intersectionnumbers relationship between primitive association schemes and association subschemes(or the quotient association schemes), we can get the intersection matrices of primitiveassociation schemes by increasing variables as little as possible. Because transpose ofevery row vector of character-multiplicity table is the common characteristic vector ofintersection matrices, we get the character-multiplicity table. Also, according to Kreinconditions of association subschemes (or quotient association schemes), Krein conditionsof primitive association schemes are reduced.