Representations of Finite Groups and Integral Lattices

这一建议将几个数学领域联系在一起：李型有限群、积分格和线性码，以及有限置换群。主要的统一成分是表象理论。PI继续研究李型有限群表示理论的一些问题，这些问题的进展将导致在积分格理论和有限本原置换群理论中的重要应用。PI建议确定李型有限群的复数表示，它是定义特征的不可约模。他还打算对低维李型有限群的交叉特征表示进行分类。然后，PI应用这两个项目的结果，在一些应用上取得了重大进展：关于全局不可约格分类的Thompson-Gross问题，寻找欧几里德积分格的最小值的问题，‘提升’问题，以及关于有限结构经典群的某些拟单子群的极大值的问题。数学中的群是由对称性的概念发展而来的。在物理、化学或数学中，对象的对称性是由一个组编码的，而这个组携带着许多关于对象本身结构的重要信息。表象理论允许人们通过群在向量空间上的作用来研究群。一个世纪以来，它一直吸引着数学家，并在物理学和化学中有许多重要的应用。因此，李群的表象理论在量子力学和基本粒子理论中起着至关重要的作用。李群的有限类似--Liettype的有限群--及其表示在编码理论和密码学中已经被证明是有价值的，并有望在计算机和通信的新时代发挥重要作用。PI的研究现在和将来主要集中在这类重要群的表示理论及其应用上。

This proposal ties together several areas of mathematics: finite groups of Lie type, integral lattices and linear codes, and finite permutation groups. The main unifying ingredient is the representation theory. The PI continues hisinvestigation on some problems on representation theory of finite groups of Lie type; progress on these problem will lead to important applications in the integral lattice theory and in the theory of finite primitive permutationgroups. The PI proposes to determine the complex representations of a finitegroup of Lie type, which are irreducible modulo the defining characteristic.He also intends to classify cross-characteristic representations of finitegroups of Lie type of low dimension. The PI then applies the results on thesetwo projects to achieve significant progress on a number of applications: theThompson-Gross problem on classifying globally irreducible lattices, the problemof finding the minimum of Euclidean integral lattices, the ``lifting'' problem,and the problem on maximality of certain quasi-simple subgroups of finiteclassical groups.The main area of research in this proposal is group theory and therepresentation theory of groups of Lie type. Groups in mathematics grew out ofthe notion of symmetry. The symmetries of an object in physics, chemistry, ormathematics, are encoded by a group, and this group carries a lot ofimportant information about the structure of the object itself. Therepresentation theory allows one to study groups via their action on vectorspaces. It has fascinated mathematicians for a century and hadmany important applications in physics and chemistry. Thus, the representationtheory of Lie groups played a vital role in quantum mechanics and in the theory of elementary particles. The finite analogs of Lie groups - finite groups of Lietype - and their representations have already proved valuable in coding theoryand cryptography, and are expected to play an important role in the new era ofcomputers and communications. The PI's research is and will be mostly focused on the representation theory of this important class of groups and its applications.