This proposal focuses on several important problems in representation theory of finite groups and its applications. Many of these problems come up naturally in the group representation theory, and others are motivated by various applications, particularly in the theory of finite primitive permutation groups, Lie algebras, integral lattices and linear codes, combinatorics, curve theory, and quantum information processing. They tie together different areas of mathematics, with the main unifying ingredient being the representation theory. The investigator intends to continue his long-term project to classify cross characteristic representations of finite groups of Lie type of low dimension. The second main project centers around certain local-global problems, which should provide links between rationality properties of complex and Brauer characters of a given finite group on the one hand and the structure of the group on the other hand. He then applies the results on these two projects to achieve significant progress in a number of applications, including the subgroup structure of finite simple groups, minimal polynomials of group elements in linear representations, integral lattices and grassmannian designs, derangements in primitive permutation groups and rational points of curves, mutually unbiased bases and quantum information processing.
The main area of research in this proposal is the representation theory of finite groups. Groups in mathematics grew out of the notion of symmetry. The symmetries of an object in nature or science are encoded by a group, and this group carries a lot of important information about the structure of the object itself. The representation theory allows one to study groups via their action on vector spaces which models the ways they arise in the real world. It has fascinated mathematicians for more than a century and had many important applications in physics and chemistry, particularly in quantum mechanics and in the theory of elementary particles. Finite groups and their representations have already proved valuable in coding theory and cryptography, and are expected to continue to play an important role in the modern world of computers and digital communications. The investigator's research will lead to important advances in understanding the representation theory of finite groups and help achieve significant progress in a number of its applications.
