Filtrations, Factorizations and Explicit Formulae for Harmonic Maps

We use filtrations of the Grassmannian model to produce explicit algebraic formulae for harmonic maps of finite uniton number from a Riemann surface to the unitary group for a general class of factorizations by unitons. We show how these specialize to give explicit formulae for harmonic maps into the special orthogonal and symplectic groups, real, complex and quaternionic Grassmannians, and the spaces SO(2m)/U(m) and Sp(n)/U(n), i.e., all the classical compact Lie groups and their inner symmetric spaces. Our methods also give explicit J2-holomorphic lifts for harmonic maps into Grassmannians and an explicit Iwasawa decomposition.