Research on representations of association schemes

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The results of this project is as follows:1. We showed that every association scheme of prime order is commutative. Moreover, if the minimal splitting field of the scheme is an abelian number field, then the scheme is algebraically isomorphic to a well-known one.2. We generalized Clifford theory of finite groups to association schemes. Especially, we found a method to construct new irreducible representations from a given one.3. We showed that, if a scheme of prime square order has a non-trivial thin closed subset, then the scheme is commutative.4. We showed that, if a scheme of prime square order has a proper strongly normal closed subset, then the scheme is commutative.5. We defined nilpotent schemes and investigated their basic properties.6. We considered modular standard modules of association schemes. Consequently, we could distinguish some algebraically isomorphic schemes by their invariants.7. We considered products of characters of association schemes and gave a sufficient condition of the product to be again a character.8. We gave an inequality for the order of a primitive association schemes. In a sense, our bound is better than the well-known "absolute hound condition.