奇异融合系的研究

The theory of fusion systems is a new theory, which was developed in recent years from finite group and its representation theory. Fusion system is widely used in finite group theory, representations of finite groups and algebraic topology. Exotic is a very important property of fusion systems. The classification of the simple fusion systems, which is closely related to the classification of the finite simple groups, proceeds by partitioning the simple fusion systems into those of realizable and those of exotic. .. The project will be devoted to the classification of the simple exotic fusion systems by constructing and studying the exotic fusion systems. The project is proposed to look for exotic fusion systems as realizable fusion subsystems, and to classify saturated fusion systems over some classes of finite p-groups, then to classify exotic fusion systems, furthermore, to find a general systematically method to construct the exotic fusion systems. We will study the properties of Burnside rings and Dade groups of fusion systems, as well as the categorical and topological properties of fusion systems, then discuss the characteristics and invariants of fusion systems, and characterize the exotic fusion systems. We will classify some classes of simple exotic fusion systems and consider the standard form problem for them. The project will enrich the theory of the exotic fusion systems and promote the classification of the simple exotic fusion systems.

融合系是近年来从有限群及其表示论中发展起来的新理论，它在有限群、有限群表示和代数拓扑等领域有着广泛的应用。奇异性是融合系中非常重要的性质，单奇异融合系的分类是单融合系分类的重要组成部分，单融合系分类与有限单群分类密切相关。. 本项目致力于通过构造和研究奇异融合系来推进单奇异融合系的分类工作。本项目拟寻找可实现融合系的奇异子系，并试图给出饱和子系是奇异的条件，以及研究实现融合系的群的性质特征，分类某些有限p-群类上的饱和融合系，分类其中的奇异融合系，考虑这些奇异融合系的标准形式问题，以寻求构造奇异融合系的一般方法。研究融合系的Burnside环和Dade群等代数性质以及融合系的范畴和拓扑性质，讨论融合系的特征性质和不变量，寻求奇异性的特征刻画。分类某些单奇异融合系并进行特征刻画。本项目的研究将丰富奇异融合系的理论并推进单奇异融合系的分类工作。

分类单融合系，特别是构造和分类单奇异融合系是当前国际前沿热点问题。本项目主要完成了A2群上饱和融合系的分类工作，得到它们是抵抗群的充分必要条件。研究了融合系的控制融合性质，特别是幂零融合系的特征刻画和超聚焦子群的交换子群在控制融合定理中具有检测意义。研究了奇异融合系在直积下的遗传性质，得到两个融合系的直积是奇异的当且仅当其中有一个是奇异融合系。在分类过程中仅用到它们的特征性质，这种处理方式使得这些技术具有一般性的可能。将控制融合定理推广到超聚焦子群情形，使之更方便有效地应用。奇异融合系的判定结论给出了直积扩张下融合系奇异性的判别办法。这些结果可能在奇异融合系的研究中发挥作用。