We establish explicit formulae for canonical factorizations of extended solutions corresponding to harmonic maps of finite uniton number into the exceptional Lie group G 2 in terms of the Grassmannian model for the group of based algebraic loops in G 2 . A description of the "Frenet frame data" for such harmonic maps is given. In particular, we show that harmonic spheres into G 2 correspond to solutions of certain algebraic systems of quadratic and cubic equations.

We classify all harmonic maps with finite uniton number from a Riemann surface into an arbitrary compact simple Lie group G, whether G has trivial centre or not, in terms of certain pieces of the Bruhat decomposition of the group Ω alg G of algebraic loops in G and corresponding canonical elements. This will allow us to give estimations for the minimal uniton number of the corresponding harmonic maps with respect to different representations and to make more explicit the relation between previous work by different authors on harmonic two-spheres in classical compact Lie groups and their inner symmetric spaces and the Morse theoretic approach to the study of such harmonic two-spheres introduced by Burstall and Guest. As an application, we will also give some explicit descriptions of harmonic spheres in low dimensional spin groups making use of spinor representations.

We classify all harmonic maps of finite uniton number from a Riemann surface into SU(n) in terms of certain pieces of the Bruhat decomposition of Ω alg SU(n), the subgroup of algebraic loops in SU(n). We give a description of the "Frenet frame data" for such harmonic maps in a given class.

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In this paper we describe how the operation of adding a uniton arises via the DPW method of obtaining harmonic maps into compact Riemannian symmetric spaces out of certain holomorphic one forms. We exploit this point of view to investigate which unitons preserve finite type property of harmonic maps. In particular, we prove that the Gauss bundle of a harmonic map of finite type into a Grassmannian is also of finite type.