Filtrations, Factorizations and Explicit Formulae for Harmonic Maps

We use filtrations of the Grassmannian model to produce explicit algebraic formulae for all harmonic maps of finite uniton number from a Riemann surface, and so all harmonic maps from the 2-sphere, to the unitary group for a general class of factorizations by unitons. We show how these specialize to give explicit formulae for such harmonic maps to each of the classical compact Lie groups and their inner symmetric spaces—the nonlinear σ-model of particle physics. Our methods also give an explicit Iwasawa decomposition of the algebraic loop group.