Factorization in unitary loop groups and reduced words in affine Weyl groups

The purpose of this dissertation is to elaborate, with specific examples and calculations, on a new refinement of triangular factorization for the loop group of a simple, compact Lie group K, first appearing in [34]. This new factorization allows us to write a smooth map from S1 into K (having a triangular factorization) as a triply infinite product of loops, each of which depends on a single complex parameter. These parameters give a set of coordinates on the loop group of K. The order of the factors in this refinement is determined by an infinite sequence of simple generators in the affine Weyl group associated to K, having certain properties. The major results of this dissertation are examples of such sequences for all the classical Weyl groups. We also produce a variation of this refinement which allows us to write smooth maps from S1 into SU( n) as products of 2n+1 infinite products. By analogy with the semisimple analog of our factorization, we suggest that this variation of the refinement has simpler combinatorics than that appearing in [34]