Representations of Finite Groups and Integral Lattices

  • Tiep, Pham P.H. (PI)

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.

StatusFinished
Effective start/end date6/1/005/31/04

Funding

  • National Science Foundation: $67,500.00

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