射影表示理论

After Specht and Macdonald’s classic works on the wreath product of a finite group and the symmetric group of n numbers, the wreath product becomes one of the important research objects of representation theory. In 1980 Macdonald constructed all irreducible modules of wreath products by polynomial functors. This categorification perfectly shows that there is an equivalence between the category of homogeneous irreducible polynomial functors of degree n and the category of finite dimensional modules of the wreath product. Recently, an important part of Schur’s classical work on projective representations of the symmetric groups has been extended into the wreath products by the applicant and her collaborator. They completed the projective character tables of wreath products...Based on the applicant and her collaborator’s work , this project aims to extend Macdonald’s categorification to the projective representations of the wreath products. More specifically, we will construct the projective modules of the wreath products by analogous “projective” (or “spin”) polynomial functors and then show that there is an equivalence between the category of homogeneous irreducible “projective” polynomial functors of degree n and the category of finite dimensional irreducible projective modules of the wreath product. This categorification will reveal the intrinsic significance of the projective representations.

对称群和任意有限群的圈积群经过Specht和Macdonald的经典工作之后成为表示论的重要研究对象之一。Macdonald在1980年提出一套新的范畴化方法，即用多项式函子构造出了圈积群的所有不可约模。Macdonald完美地得到了圈积群的有限维不可约模与度为n的齐次不可约多项式函子之间的等价关系。最近项目申请人和合作者把Schur在对称群射影表示的经典工作上的一个重要部分成功推广到圈积群上，从而补充完善了圈积群的不可约射影特征标表。结合申请人和合作者关于圈积群射影表示的研究，本项目主要研究计划是推广Macdonald的范畴化方法，构造新的“射影”多项式函子来研究圈积群的射影表示，证明射影多项式函子范畴与圈积群的双重覆盖群的有限维不可约模范畴之间的等价性。通过具体实现圈积群的Macdonald范畴化，来深刻揭示射影表示的内在意义。

在Schur经典的对称群的射影理论基础上，我们确定了Hyperoctahedral群与任意有限群的圈积群的射影特征标值，证明了此圈积群由奇划分函数确定的自旋特征在由奇划分函数确定的共轭类上的取值大部分为零，而只在一类被着色了的共轭类上取值不为零。这项工作完善了有限群的射影表示理论。另外，根据Macdonald的范畴化方法，我们已经找到了用strict划分确定的这类不可约“spin”多项式函子来构造对称群的有限维不可约射影模。这对后续圈积群的范畴化工作的继续开展起到了关键性作用。