Investigations on the conjectures of McKay and Alperin-McKay

提出的研究项目是在有限群的表示理论。我们的目的是调查两个突出的和长期存在的开放猜想，麦凯猜想和它的细化由Alperin， Alperin-McKay猜想，分别。这些猜想是在上世纪70年代中期形成的。在最初的表述中，麦凯猜想假设两个看似不相关的对象集合具有相同数量的元素，这是由群的表征理论构建的。其中一组由“本地数据”定义，即由适当的子组构造的数据定义，另一组仅由组本身的“全局数据”定义。全局数据应该由局部数据决定，这是麦凯猜想和其他著名的表征理论猜想背后的哲学。最近的两个事态发展激发了这个项目的灵感。首先，在2007年，Isaacs， Malle和Navarro发表了McKay猜想的一个强大的约简定理。这就留给我们去验证有限单群的一些相当复杂的条件。其次，Späth 2007年博士论文的结果提供了一种方法来验证这些复杂的条件在当地的情况下，至少在特殊情况下。

The proposed research project is located in the representation theory of finite groups. Our aim is the investigation of two prominent and long standing open conjectures, the McKay conjecture and its refinement by Alperin, the Alperin-McKay conjecture, respectively. These conjectures were formulated in the mid 70th of the last century. In its original formulation, the McKay conjecture postulates that two seemingly unrelated sets of objects, constructed from the representation theory of a group, have the same number of elements. One of these sets is defined by “local data”, i.e. by data constructed from proper subgroups, the other set only by “global data” of the group itself. That global data should be determined by local data, is the philosophy behind the McKay conjecture and other famous conjectures of representation theory. Two recent developments have inspired the proposed project. Firstly, in 2007, Isaacs, Malle and Navarro published a powerful reduction theorem for the McKay conjecture. This leaves one to verify some rather complicated conditions for the finite simple groups. Secondly, the results of Späth’s 2007 PhD thesis provide a means to verify these complicated conditions in the local situation, at least in special cases.