This is Rupert Venzke's webpage for the Summer 2002 DIMACS REU Program.

Program & People

This summer, I will be working on the third VIGRE Project: Representation Theory. My advising professors are Dr. Shawn Robinson and Dr. Friedrich Knop. Graduate students Bill Cuckler and Vince Vatter will also be assisting us in the research.

We will be concerned with the Representation Theory of both Lie Groups and Lie Algebras.

an Introduction

We begin with a few preliminary definitions:

First of all, a Lie group is essentially a smooth manifold equipped with some additional group structure. This group structure requires that the maps x → g o x and x → x^-1 both be continuous. By a smooth manifold, we roughly mean an (infinitely) differentiable, locally Euclidean Hausdorff space that is "second countable." In other words, the space under consideration has just about all of the properties one would normally expect of a smooth surface in R^3.

One class of Lie Groups is SL(n, F), the group of n x n matrices over a field F, with determinant 1. These objects have manifold structure, since they can be interpreted as zero sets of the polynomial det(M) - 1. We know that the structure is smooth, since multi-variate polynomials are differentiable. The algebraic operation is the usual matrix multiplication. The map x → g o x is continuous, since it amounts to the evaluation of a polynomial. We also see the map x → x^-1 is continuous by utilizing the classical adjoint matrix inverse formula.

Two other important classes are O(n, F), the group of n x n matrices A over F satisfying A o (A^T) = I, and SO(n, F), the subgroup of O(n), consisting of those matrices which additionally have determinant 1. The Lie Group O(n) can naturally be interpreted as those length-preserving linear transformations of an n-dimensional vector space, while SO(n) can be interpreted as those linear transformations that preserve both length and orientation.

Naturally enough, these Lie Groups (as well as several others) arise often in physics.

As a more explicit example, consider the unit circle in the complex numbers, with elements {e^{i t}}, where t varies from 0 to 2π. This inherits a naturally induced metric topology and is easily seen to be differentiable. Further, it has the additional algebraic structure given by e^z e^w = e^{z + w}. The map x → g o x is continuous, since it can be interpreted as a rotation by some number of radians. The map x → x^-1 is also continuous since it can be thought of as a reflection across the x-axis.

the Purpose of Representations

What is a representation in this context? A representation of a Lie Group G is basically a structure-preserving map from G into GL(n), the general linear group of n x n matrices over F (matrices with non-zero determinant).

What we are doing here is associating matrices with group elements. The idea behind Representation Theory is to transform inquiries about abstract objects into better understood questions in Linear Algebra. We are interested in the characters of these groups.

Specifically, a character χ on the group G, with representation φ, is the map χ: G → F defined by χ(g) = Trace(φ(g)).

Why characters? A representation of a Lie Group is called irreducible if its corresponding group-ring module has no non-trivial submodules under the action of FG. A fundamental result in the theory is that, under certain standard conditions, a representation of a Lie Group can be decomposed into a direct sum of irreducibles, involving non-negative integral coefficients. We would like to distinguish between these irreducibles and find these coefficients. As it turns out, we can develop a bijection between the irreducible decompositions and characters.

Classically, one can compute all of these characters using combinatorial formulas. However, these formulas can often be difficult to work with. Consequently, we would like some way to visualize the decomposition of these spaces.

To do this, we first reformulate the problem in terms of Lie Algebras. Given any Lie Group with representation ρ, we associate a Lie Algebra with the induced representation. Our Lie Algebra will simply be the tangent space to the Lie Group at the identity, and its representation will be the differential of ρ mapping it into the module of endomorphisms on a vector space.

More generally, a Lie Algebra is essentially a vector space with a bracket product and satisfying certain important relations. However, we will be concerned with just those Lie Algebras which occur as tangent spaces to Lie Groups. The whole idea here is that, in this context, Lie Algebras are much easier to understand. Once we get results in this theory, we then convert those results back into useful statements in the representation theory of Lie Groups.

Recent Progress in the Theory

Recently, a "path model" has been developed by Peter Littelmann ('95) to visualize the corresponding decomposition of Lie Algebras. This characterization identifies irreducibles with certain points in a rational lattice. Finding our coefficients then amounts to counting the number of certain paths in the lattice joining the origin with such a specified point.

We have implemented such a model for rank 2 systems as an applet - one that allows us to easily visualize the elusive coefficients determining such decompositions.