Fusion Schemes of Semidirect Products of Association Schemes

Let X = (X, H) and Y = (Y, K) be two association schemes. In [1], an (external) semidirect product Z = (X × Y, H nπ K) of Y and X relative to π is defined. In this paper, given a normal closed subset N of K which is fixed by π, we find a way to construct a fusion scheme of Z with respect to N . We then show that every association scheme obtained by any of three products, direct, wreath and semidirect product, of X = (X, H) and Y = (Y, K), is realized as a fusion scheme of Z relative to a proper π with respect to a suitable normal closed subset N of K. Finally, we give some examples of the fusion schemes of Z with respect to a certain N , which are not decomposed into their subschemes via any of the three product operations.