Representations of Finite Groups and Integral Lattices

University Of Florida, Gainesville FL

Investigators

Abstract

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 his investigation 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 permutation groups. The PI proposes to determine the complex representations of a finite group of Lie type, which are irreducible modulo the defining characteristic. He also intends to classify cross-characteristic representations of finite groups of Lie type of low dimension. The PI then applies the results on these two projects to achieve significant progress on a number of applications: the Thompson-Gross problem on classifying globally irreducible lattices, the problem of finding the minimum of Euclidean integral lattices, the ``lifting'' problem, and the problem on maximality of certain quasi-simple subgroups of finite classical groups. The main area of research in this proposal is group theory and the representation theory of groups of Lie type. Groups in mathematics grew out of the notion of symmetry. The symmetries of an object in physics, chemistry, or mathematics, 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. It has fascinated mathematicians for a century and had many important applications in physics and chemistry. Thus, the representation theory 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 Lie type - and their representations have already proved valuable in coding theory and cryptography, and are expected to play an important role in the new era of computers 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.