Representations of Finite Groups and Integral Lattices

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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.