Computing Isomorphisms of Association Schemes and its Application

Using computers, Hanaki and I were classifying association schemes of small order for a past few years (Hanaki and Miyamoto (1998,1998,2000),Hanaki (1999)). In the present paper we will discuss about computing isomorphisms of association schemes. We compute isomorphisms of an association scheme using its relation matrix defined in the following section. A relation matrix can be seen an adjacency matrix of a labeled graph. If we consider an isomorphism from an association scheme to itself, the isomorphism may be given by a permutation on the set of vertices of the labeled graph. In our case we generally consider that an isomorphism may permute also the labels of the graph. The most familiar examples of association schemes are given by transitive permutation groups on the set of vertices. Then an element of the transitive group is an isomorphism of the former type. An element of the normalizer of the group is also an isomorphism and it may permute the labels. The author gives an algorithm to speed up to compute normalizers of permutation groups as an application of computing isomorphisms of association schemes in Miyamoto (2000). In the present paper we will also give some more inspection about this algorithm. In the following section we will define association schemes and will introduce necessary properties. For more basic properties, readers may refer to Bannai and Ito (1984). For graph isomorphisms and computing normalizers readers may also refer to Mckay (1981) and Theißen (1997). A program computing isomorphisms of labeled graphs is not familiar and if we do not consider the algebraic property of association schemes, we have to compute combinatorially large numbers of permutations on the labels in some graphs. In fact our old program was sufficient to compute an isomorphism from one association scheme to another in the classification of the isomorphism classes stated above but sometimes it took too much time to compute all the isomorphisms of an association scheme to itself. The labeled graph given by an association scheme is regular with respect to each label. This restricts the permutations of isomorphisms on the labels. If an association scheme is defined by a transitive group