REPRESENTATION THEORY AND COMBINATORICS AND RELATED TOPICS

Koike在正交群的自然表示和基本自旋表示的张量空间中构造了自旋群的所有不可约表示。这种结构是H.Weyl在他的名著《古典群体》中提出的论点的自然延伸，并弥补了缺失的情况。即，Koike考虑了上述张量空间中自旋群的中心化代数（这些代数是经典情况下对称群和Brauer中心化代数的自然类似物），并给出了上述代数的显式基，这些基由“扩展Brauer图”参数化。Koike还定义了上述张量空间的子空间，其中对称群和自旋群充当对偶对。在合作论文中，Taniguchi将Kawanaka Noriaki（大阪大学）在对称群中引入的一类Molien级数（它是一个单变量q的有理多项式）的概念推广到所有有限反射群。谷口和他的合作者给出了这些有理多项式的明确公式，并揭示了Iwahori Hecke代数的双面细胞与这些多项式的联系。Yamaguchi给出了I.Schur经典结果的表征背景，给出了对称群的不可约射影表示的特征。即基于超八面体群的扭转群代数与李超代数的Sergeev对偶性，Yamaguchi定义了对称群的扭转群代数在上述扭转群代数中的嵌入，并给出了对称群的扭转群代数与李超代数作为对偶对的子空间。

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.