Representation Theory, Combinatorics and Related Topics

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For the irreducible representations of the double covering groups Spin(N) of the special orthogonal groups SO(N), Koike gave natural generalization of Weyl's construction developed in his famous book 'The Classical Groups'. In the classical cases, the centralizer algebras of tine classical groups in the tensor product spaces of the natural representation play crucial roles. Koike showed that similar arguments hold for the Spin groups and described the centralizer algebras of the Spin groups and picked up all the irreducible representations of the Spin groups. It made the reason clear why apparently complicated identities such as Ramanujan's sum formulas etc hold and showed that the symmetries under the Weyl groups (or the root systems) are essential for them to hold. From the above view point, he gave consistent explanations of several identities which are found independently and further he found new identities, which gives the infinite product formulas of Jackson integral for root system F 4.