Algebra VIGRE Group

Dave Benson, Brian Boe, and Dan Nakano

Conjugacy Classes via Cohomology and Representation Theory (i.e. jazzed up Jordan Canonical Forms).

The main goal of the project is to study conjugacy classes of matrices using new and modern machinery via representation theory and cohomology. The prerequisites are a good understanding of linear algebra and group theory. We will begin by looking at very concrete open problems involving matrices based on work of Carlson, Lin, Nakano and Parshall. This entails calculating the restricted nullcone (i.e. matrices when multiplied p times gives you zero) for low primes. If time permit other open problems will be to

1) calculate the support varieties of Weyl modules for small primes.

2) calculate the support varieties for the Specht modules for symmetric groups.

At various stages during this project, lectures will be given to introduce students to the connections between these concrete calculations the underlying modern machinery involving Lie/group theory, cohomology and representation theory.

Introductory lecture Monday, August 18, 2:30, by Dan Nakano:

Abstract: In this introductory talk I will present the basic background material concerning the orbit theory for the general linear group.The orbit theory has a deep and rich structure with connections to both cohomology and representation theory.The main goal of this VIGRE seminar will be to study the structures of certain varieties of nilpotent matrices called restricted nullcones.There are many open questions about the structures of these varieties. It is hoped that progress by the VIGRE group can be made in determining these varieties when the prime of the underlying field has small characteristic.