Semidirect Products of Association Schemes

In his 1996 work developing the theory of association schemes as a ‘generalized’ group theory, Zieschang introduced the concept of the semidirect product as a possible product operation of certain association schemes. In this paper we extend the semidirect product operation into the entire set of association schemes. We then derive a way to decompose certain association schemes into smaller association schemes. We also investigate to what extent this product helps us to understand and characterize the structure of association schemes. We give some examples to show that the semidirect product produces many schemes that cannot be described as neither the direct product nor the wreath product of smaller schemes.