  We give a completely explicit formula for all harmonic maps of finite uniton
number from a Riemann surface to the unitary group U(n) in any dimension, and
so all harmonic maps from the 2-sphere, in terms of freely chosen meromorphic
functions on the surface and their derivatives, using only combinations of
projections and avoiding the usual dbar-problems or loop group factorizations.
We interpret our constructions using Segal's Grassmannian model, giving an
explicit factorization of the algebraic loop group, and showing how to obtain
harmonic maps into a Grassmannian.
