RUI: Modular Representations of Algebraic Groups and Related Structures

University Of Wisconsin-Stout, Menomonie WI

Investigators

Abstract

Abstract This project is an investigation of the representation theory of algebraic groups, certain finite subgroups, and associated Lie algebras over fields of prime characteristic. Over prime fields, the fundamental open questions involve cohomology and extensions of modules, and such questions are the focus of the project. Of particular interest are two one-parameter families of subgroups associated to a reductive algebraic group - finite Chevalley groups and Frobenius kernels. Recent results have uncovered deep relationships between these two theories, and an underlying goal of the project is to strengthen these connections. Such relationships are of inherent theoretical interest and can be powerful computational tools for transferring information from one category to the other as well as to other areas. Professor Bendel in partial collaboration with Daniel Nakano and Cornelius Pillen will build upon previous work of the trio to make computations of cohomology and extensions groups as well as support varieties, while seeking to extend known relationships. Computations of ordinary Lie algebra cohomology for small primes, which have independent interest, will be used to meet these objectives. Algebraic structures such as groups and Lie algebras arise naturally in a number of physical situations, for example in the symmetries found in natural phenomenon, and have been applied in biology, physics, chemistry, and computer science. In basic terms, a representation of a group (or Lie algebra) provides a geometric glimpse at the structure of the group. A typical group has many representations of varying sizes. By better understanding the representations of a group, one can better understand the group and extract information for applications. One of the main objectives of this work is to investigate the fundamental question of how different representations can be combined to form new representations. The principal investigator has been actively involved in promoting new mathematical ideas. He is currently organizing a conference that relates representation theory to other algebraic and geometric areas. At the local level, another objective of this project is to enhance the teaching and learning environment of the Mathematics, Statistics, and Computer Science Department at the University of Wisconsin-Stout (a predominantly undergraduate institution) and to increase student interest in scientific research. Undergraduate students will be directly involved in the project by making computer calculations, requiring them to combine their mathematical and programming skills. Further, the project will promote the development of a research climate within the department where faculty bring fresh ideas into the classroom and students, especially women, are regularly exposed to mathematics beyond the classroom and have the opportunity to participate in faculty research.