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 componentwise. 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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