Project Details
Description
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.
Status | Finished |
---|---|
Effective start/end date | 6/1/00 → 5/31/04 |
Funding
- National Science Foundation: $67,500.00