Sungyell Song:
Fusion schemes of semidirect products of association schemes.

We can construct new association schemes by taking direct product and wreath product of two association schemes. For these operations we need not put many restrictions on the association relations of the association schemes we are combining because the association relations for the products are defined using the association relations of the factors component-wise. Another more subtle way to construct association schemes is by the semidirect product. More interesting than constructing arbitrary external direct or semidirect products is discovering which association schemes are internal direct or semidirect products of two (or more) of their subschemes. As we study more on the semidirect product, we are able to better understand the structures of certain association schemes of small order.

It is easy to see that the semidirect product relative to a trivial homomorphism is the direct product of the two association schemes as in the theory of groups. We have recently found a way to construct new association schemes as fusion schemes of the semidirect product scheme. This composition is a unified and generalized form of all three product operations; in the sense that every product association scheme obtained via any of the three products can be interpreted as a fusion scheme of the semidirect product relative to a certain homomorphism. Our aim is to illustrate how to make use of the semidirect product operation and the fusion process in the study of characterization and classification problems of association schemes among others.

We will begin by reviewing some preliminary material including some properties of three product operations of association schemes. Then we will describe a composition of association schemes using the fusion process and the semidirect product operation. We then show how this process produces various association schemes that can be obtained by taking other product operations. As time permits, we will see some critical examples which are decomposed by the above process but not by any of the three product operations mentioned. (This talk is based on joint work with Sejeong Bang and Mitsugu Hirasaka in Korea.)

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