REPRESENTATION THEORY AND COMBINATORICS AND RELATED TOPICS

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Koike constructed all the irreducible representations of the Spin groups in the tensor spaces of the natural representations of the orthogonal groups and the fundamental Spin representation. This construction is a natural extension of the argument which is developed by H.Weyl in his famous book "The Classical Groups" and makes up the missing cases. Namely Koike considers the centralizer algebras of the Spin groups in the above tensor spaces (These algebras are natural analogs of the symmetric groups and Brauer's centralizer algebras in the classical cases.) and gives the explicit bases of the above algebras which are parameterized by the "extended Brauer diagrams". Koike also defines the subspaces of the above tensor spaces on which the symmetric group and the Spin group acts as the dual pair.In collaborated papers, Taniguchi extends the notion of a kind of the Molien series (it is a rational polynomial in one variable q.) to all the finite reflection groups, which was introduced by Kawanaka Noriaki (Osaka University) in case of the symmetric groups. Taniguchi and his collaborators give explicit formulas of those rational polynomials and reveal connections of the two-sided cells of the Iwahori Hecke algebras and those polynomials.Yamaguchi shows the representational background of the classical result of I.Schur which gives the characters of the irreducible projective representations of the symmetric groups. Namely based on Sergeev's duality of the twisted groups algebras of the hyperoctahedral groups and the Lie superalgebras, Yamaguchi defines immersions of the twisted groups algebras of the symmetric groups into the above twisted groups algebras and gives the subspaces, on which the twisted groups algebras of the symmetric groups and the Lie superalgebras act as the dual pair.