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We use filtrations of the Grassmannian model to produce explicit algebraicformulae 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 generalclass of factorizations by unitons. We show how these specialize to giveexplicit formulae for such harmonic maps to each of the classical compact Liegroups and their inner symmetric spaces - the nonlinear sigma-model of particlephysics. Our methods also give an explicit Iwasawa decomposition of thealgebraic loop group.

Filtrations, factorizations and explicit formulae for harmonic maps